mathematics formulas and universal constants

FORMULAS—BEAUTIFUL FORMULAS, SCIENTIFIC FORMULAS, UNIVERSAL PHYSICAL CONSTANTS

John hebert  5/13/18

CONTENTS—

Beautiful mathematics formulas

Scientific formulas

Universal physical constants

BEAUTIFUL FORMULAS

8/29/17-8/30/17;9/15/17;11/21/17-11/22/17;12/25/17; 4/19/18

CONTENTS

  1. Dirac’s equation
  2. Einstein’s field equation
  3. Maxwell’s equations
  4. General relativity
  5. Special relativity
  6. Schrodinger’s equation
  7. Uncertainty principle
  8. Gibb’s statistical mechanics
  9. Stephan-Boltzmann law
  10. e=mc^2
  11. Laplace equation
  12. De broglie relation-matter wave
  13. Navier-stokes equations
  14. Riemann zeta function
  15. Noether theorem
  16. Euler-lagrange equation
  17. Hamilton quanternion
  18. Standard model
  19. Lagrange formula
  20. cantor inequality
  21. Riemann hypothesis
  22. Hawking-Bekenstein entropy formula
  23. Heat equation
  24. wave equation
  25. poisson equation
  26. Wave-particle duality
  27. fundamental theorem of calculus
  28. Pythagorean theorem
  29. Gauss-Bonnet theorem
  30. universal law of gravitation
  31. Newton’s 2nd law of motion
  32. kinetic energy
  33. Potential energy
  34. 2nd law of thermodynamics
  35. principle of least action
  36. Spherical harmonics
  37. Cauchy residue theorem
  38. Callen-Symanzik equation
  39. Minimal surface equation
  40. Euler 9 point center
  41. Mandelbrot set
  42. Yang-Baxter equation
  43. Divergence theorem
  44. Baye’s theorem
  45. logistic map
  46. Einstein’s law of velocity addition
  47. Photoelectric effect formula
  48. Faraday law
  49. Cauchy momentum equation
  50. De moivre’s theorem
  51. Fourier transform
  52. prime counting function
  53. Murphy’s law
  54. Summation formula
  55. Logarithmic spiral
  56. Heron’s formula
  57. Quadratic equation
  58. Euler line
  59. Pythagorean triple formula
  60. Euler’s formula
  61. Simplex method
  62. Proof of infinity of prime numbers
  63. Harmonic series
  64. Euler sums
  65. Cubic equation
  66. Quartic equation
  67. quintic equation
  68. Lorentz equation
  69. Euler-lagrange formula
  70. Euler product formula
  71. Euler-maclaurin formula
  72. Pi
  73. Exponent
  74. Natural logarithm
  75. Conic sections
  76. exponential growth or decay
  77. Calculation an orbit I.e. a comet
  78. interesting number idea 1
  79. interesting number idea 2
  80. interesting number idea 3

EQUATIONS

  1. 1. Dirac equation 

The Dirac differential equation from quantum mechanics predicted the existence of antimatter in 1928. Antimatter are particle of the same mass and spin, but have an opposite charges than their counterparts of matter.

2. Einstein field equation

The einstein field equations, or the einstein-hilbert equations is used to describe gravity classically. It uses geometry to model gravity’s effects.

3. Maxwell’s equations

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James clerk maxwell formulated 4  differential equations which describes how charged particles produce an electric and magnetic force. They calculate motion of particles in electric and magnetic fields, and describe how electric charges and electric currents create electric and magnetic fields, and vice versa. The 1st equation can be used to calculate the electric field produced by a charge.

The 2nd equation  calculates the magnetic field.  The 3rd equation, called ampere’s law, shows how the magnetic fields circulate around electric currents and time varying electric fields. The 4th equation, named Faraday’s law, describes how the electric fields circulate around time varying magnetic fields.

4.  General relativity

In 1915, Albert einstein formulated the general theory of relativity. It deals with space and time, two aspects of spacetime. Spacetime curves whenever there are gravity, matter, energy, and momentum. The general theory of relativity is centrally about the principle of equivalence. General relativity shows that light curves in an accelerating frame of reference. It also says that light will bend and it will slow down in the presence of  massive amounts of matter.

5. SPECIAL RELATIVITY

The Lorentz Transformations is central to the special theory of relativity and forms its mathematical basis. The special theory of relativity says that the speed of light is the same regardless of the speed the observer travels. It also describes what is relative and what is absolute about time, space, and motion. It goes on to further calculate mass increase, length shrinkage, and how much time slows down for objects moving close to the speed of light, and that a person traveling close to the speed of light would age less than that of a stationary person.

6. Schrodinger’s equation

a differential equation that is the foundation of quantum mechanics. It is one of the most accurate theories of how subatomic particles behave. This equation describes a wave function of a particle or group of particles that have a certain value at every point in space for every given time. this function contains all the information that can be known about a particle or system. The wave function gives physical properties such as position, momentum, energy, etc as real values.

7. Uncertainty principle

This principle says that trying to measure a thing’s definite position will make its momentum less known, and vice-versa.

8. GIBB’S STATISTICAL MECHANICS

Statistical mechanics, a branch of theoretical physics, uses probability theory to analyze the average behavior of a mechanical system wherein the state of the system is uncertain.

Statistical mechanics is commonly used to describe the thermodynamic behavior of large systems.

9. Stefan Boltzmann law

R=σT4

where σ is the Stefan-Boltzmann constant, which is equal to 5.670 373(21) x 10-8 W m-2 K-4, and where R is the energy radiated per unit surface area and per unit time. T is temperature, which is measured in kelvin scale. this law is only usable for the energy radiated by blackbodies but can still be used. In quantum physics, the Stefan-Boltzmann law (sometimes known as Stefan’s Law) states that the black body’s radiation energy emitted from an object is directly proportional to the temperature of the object raised to the fourth power.

10. Mass energy equivalence

E=mc^2

In physics, mass energy equivalence says that any matter’s mass has an equivalent amount of energy and vice versa. these quantities are directly related to each another.

11. Laplace’s equation

In mathematics, Laplace’s equation is a second-order partial differential equation. the harmonic functions are the solutions of Laplace’s equation, which are important in many fields of science, particularly in the fields of electromagnetism, astronomy, and fluid dynamics, where they are used to accurately describe the behavior of electric, gravitational, and fluid potentials. the Laplace equation is the steady-state heat equation when studying heat conduction.

12. DEBROGLIE RELATION/Matter wave

λ=h/mv

Where λ is the wavelength of the object, h is Planck’s constant, m is the mass of the object, and v is the velocity of the object. A correct alternate version of the formula is

λ=h/p

Where p is the momentum. (Momentum equals mass times velocity). These equations asserts that matter exhibits a particle-like nature in some circumstances, and a wave-like characteristic at other times.

13. Navier Stokes equations

The Navier Stokes equations describe the motion of fluids. It results from applying newton’s 2nd law to fluid dynamics.  They are very useful because they describe the physics of many things. They may be used to model weather, ocean currents, water flow in a pipe, the air’s flow around a wing, and the motion of stars inside a galaxy. The Navier Stokes equations also help with the design of aircraft and cars, the study of blood flow, the analysis of pollution, the design of power stations,  and many other things. Along with Maxwell’s equations they can be used to model and study magnetohydrodynamics. 

14. Riemann zeta function

ζ(s)=∑n=1 to ∞ 1ns, Re(s)>1.

Where

Re(s) is the real part of the complex numbers. For example, if s=a+ib, then Re(s)=a. (where i^2=â-1)

Riemann zeta function ζ(s) in the complex plane. In mathematics, the Riemann zeta function, a very important function in number theory because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics. The riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function. the Riemann hypothesis is considered by many mathematicians to be the most important unsolved problem in pure mathematics.

15. Noether’s theorem

dX/dt=0

Emmy noether was an influential mathematician known for her landmark contributions to abstract algebra and theoretical physics.

Noether’s theorem can be stated as follows:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.

A more complex version of the theorem involving fields states that:

To every differentiable symmetry generated by local actions, there corresponds a  conserved current.

16. Euler Lagrange equation

In the calculus of variations, the Euler Lagrange equation, is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. the Euler Lagrange equation is useful for solving optimization problems. 

17. Quaternion

a + bi + cj + dk

where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units.

In mathematics, the quaternions are a number system which extend to the complex numbers, and are applied to in 3-dimensional space. Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving 3-dimensional rotations such as in 3-dimensional computer graphics, computer vision and crystallographic texture analysis. 

18. Standard Model (mathematical formulation) for particle physics

19. LAGRANGE FORMULA

Lagrangian mechanics reformulates classical mechanics. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations. There are 2 forms of the equation- the Lagrange equations of the first kind,  and the Lagrange equations of the second kind.

20. CANTOR’S INEQUALITY/Cantor’s theorem

In elementary set theory, Cantor’s theorem is a basic result which states that, for any set A, the set of all subsets of A (the power sets of A, 𝒫(A)) has a strictly greater cardinality than A itself. the theorem implies that there is no largest cardinal number (colloquially, “there’s no largest infinity”

21. Riemann hypothesis

The Riemann hypothesis is a mathematical conjecture.  finding a proof of the hypothesis is considered by many mathematicians as one of the hardest and most important unsolved problems of pure mathematics. It is about a special function, the riemann zeta function. This function inputs and outputs complex numbers values. The inputs which give outputs of zero are called zeros of the zeta function. Many zeros have been found. And the “obvious” ones to find are the negative even integers. More have been computed and have real part 1/2. The hypothesis states all the undiscovered zeros must have real part 1/2. The functional equation also says all zeros (except the “obvious” ones) must be in the critical strip: real part is between 0 and 1.  If proven, it would allow mathematicians to better describe how the prime numbers are placed among whole numbers. The Riemann hypothesis is so important and difficult to prove that the Clay Mathematics Institute has offered $1,000,000 to the first person to prove it.

22. HAWKING-BEKENSTEIN ENTROPY FORMULA

black-hole thermodynamics studies how to reconcile the laws of thermodynamics with the existence of Black-hole event horizons. It is an effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle. The 2nd law of thermodynamics requires black holes to have entropy. If black holes had no entropy, then it would be possible to violate the second law of thermodynamics by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was absorbed by the black hole.

23. HEAT EQUATION

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. The heat equation has basic importance in a range of scientific fields. In mathematics, such as probability theory and partial differential equations.

24. Wave equation

iℏ∂/∂tΨ(x,t)=H^Ψ(x,t)

where i is the imaginary number, ψ (x,t) is the wave function, ħ is the reduced planck constant, t is time, x is position in space, Ĥ is a mathematical object known as the Hamilton operator. Equations that describe waves as they occur in nature are called wave equations. Waves as they occur in rivers, lakes, and oceans are similar to those of sound and light. The problems  describing waves come up in fields like acoustics, electromagnetic, and fluid dynamics.  the problem of a vibrating string such as that of a musical instruments was studied. In quantum mechanics, the Wave function describes the probability of finding an electron somewhere in its matter wave. The wave function concept was first introduced in the schrodinger equation.

25. Poisson’s equation

∇^2φ=f.

(∂^2/∂x^2+∂^2/∂y^2+∂^2/∂z^2)φ(x,y,z)=f(x,y,z).

When f=0 identically we obtain laplace’s equation.

Poisson’s equation is a partial differential equation of elliptic type with a wide range of utility in mechanical engineering and theoretical physics. It arises, for example, in the description of the potential field caused by a given charge or mass density distribution; when the potential field is known, it allows calculation of the  gravitational or electrostatic field.  It is a generalization of laplace’s equation, which is also frequently seen in physics. Poisson’s equation may be solved using a green’s function.

26.Wave particle duality

in the 1700s and 1800s, there was a big argument among physicists about whether light was made of particles, or waves.  Light seems to be both.  Until the 20th century, most physicists thought that light was either one or the other, and that the scientists on the opposite side of the argument were wrong. Wave particle duality means that all particles show both wave and particle properties. This is a fundamental concept of quantum mechanics. Classical concepts of “particle” and “wave” do not completely describe the behavior of quantum-scale objects.

27. Fundamental theorem of calculus

The fundamental theorem of calculus is fundamental in the study of calculus. This theorem shows the relationship between the derivative and the integral and between the definite integral and the indefinite integral. It has two parts, the first fundamental theorem of calculus and the second fundamental theorem of calculus. It asserts that the derivative and integral as inverse processes.

28. Pythagorean theorem

In mathematics, the Pythagorean theorem, is a statement about the sides of a right triangle. The Pythagorean theorem says that the area of a square on the hypotenuse (longest side of the triangle) is equal to the sum of the areas of the squares on the legs. 

a^2+b^2=c^2.

29. Gauss Bonnet theorem

The Gauss Bonnet theorem or Gauss Bonnet formula in differential geometry is an important assertion about surfaces which connects their geometry (curvature) to their topology (the euler characteristic).

30. Newton’s law of universal gravitation

Fg=Gm1m2/r2,

Newton’s universal law of gravitation is a physical law that describes the attraction between two objects with mass.

In this equation:

 Fg is the total gravitational force between the two objects.

G is the gravitational constant.

m1 is the mass of the first object.

m2 is the mass of the second object.

r is the distance between the centers of the objects.

In SI units, Fg is measured in newtons (N), m1 and m2 in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6.674×10-11 N m2 kg

31. Newton’s 2nd law of motion

F=ma.

For a particle of mass m, the net force F on the particle is equal to the mass m times the particle’s acceleration a.

32. Kinetic energy

Kinetic energy is the energy that an object has because of its motion. 

KE (joules)=(mass x velocity^2)/2

33. Potential energy

Potential energy is the energy that an object has because of its position on a gradient of potential energy called a potential field.

PE=gravity (9.81) x height x mass

34. Second law of thermodynamics

S (prime)-S>=0

The second law of thermodynamics says that when energy changes from one form to another form, or matter moves freely, entropy (disorder) increases, in a closed system.

35. Principle of least action

The principle of least action  can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive newtonian,  lagrangian and hamiltonian equations of motion, and even general relativity. The principle remains the focus in modern physics and mathematics, with applications in thermodynamics, fluid mechanics, the theory of relativity, mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in morse theory.  maupertuis principle and Hamilton’s principle are prime examples of the principle of stationary action.

36. SPHERICAL HARMONICS

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often used to solve partial differential equations that commonly occur in science. 

37. Cauchy Residue theorem

In complex analysis, the residue theorem,  is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well. It generalizes the cauchy integral theorem and cauchy integral formula. From a geometrical perspective, it is a special case of the generalized stoke’s theorem.

38. Callan Symanzik equation

In physics, the Callan Symanzik equation is a differential equation describing the evolution of the n-point correlation functions under variation of the energy scale at which the theory is defined. It involves the beta-function of the theory and the anomalous dimensions.

39. MINIMAL SURFACE EQUATION

In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature.

40. EULER’S 9 POINT CENTER/Nine-point center

In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It gets its name because is the center of the 9-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices.

41   MANDELBROT SETS

The Mandelbrot set is an important example of a fractals in mathematics.The Mandelbrot set is essential for understanding chaos theory. 

42. Yang Baxter equation

In physics, the Yang Baxter equation (or star-triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states.

43. DIVERGENCE THEOREM

In vector calculus, the divergence theorem, also known as Gauss’s theorem or Ostrogradsky’s theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.  it states that the sum of all sources (with sinks regarded as negative sources) gives the net flux out of a region. The divergence theorem is an important result for the mathematics of physics and engineering, especially in electrostatics and fluid dynamics. In physics and engineering, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to green’s theorem. The theorem is a special case of the more general stoke’s theorem.

44. Bayes’ theorem

P(A|B)=P(B|A)P(A)P(B).

In probability theory, Bayes’ theorem shows the relation between a conditional probability and its reverse form. For example, the probability of a hypothesis given some observed pieces of evidence and the probability of that evidence given the hypothesis.

45. Logistic map

xn+1=rxn(1-xn)

where xn is a number between zero and one that represents the ratio of existing population to the maximum possible population.

46. EINSTEIN’S LAW OF VELOCITY ADDITION/Velocity-addition formula

In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as thomas procession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost. Standard applications of velocity-addition formulas include the doppler shift, doppler navigation, the aberration of light, and the dragging of light in moving water. It was observed by galilei that a person on a uniformly moving ship has the impression of being at rest and sees a heavy body falling vertically downward. This observation is now regarded as the first clear statement of the principle of mechanical relativity. The cosmos of Galileo consists of absolute space and time and the addition of velocities corresponds to composition of galilean transformations. The relativity principle is called galilean relativity. It is obeyed by newtonian mechanics. According to the theory of special relativity, the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed. The cosmos of special relativity consists of Minkowski spacetime and the addition of velocities corresponds to composition of lorentz transformations. In the special theory of relativity Newtonian mechanics is modified into relativistic mechanics.

47. PHOTOELECTRIC EFFECT FORMULA

The photoelectric equation involves; h = the Planck constant 6.63 x 10-34 J s. f = the frequency of the incident light in hertz (Hz) … Ek = the maximum kinetic energy of the emitted electrons in joules (J)

The photoelectric effect refers to the emission of electrons or other free carriers when light is shone onto a material. Electrons emitted can be called photo electrons. It is commonly studied in electronic physics, as well as in fields of chemistry, such as quatuum chemistry or electrochemistry.

48. Faraday’s law of induction

Faraday’s law of induction is one of the basic laws of electromagnetism. The law explains the operation principles of generators, transformers and electric motors.

49. Cauchy momentum equation

The Cauchy momentum equation is a vector partial differential equation formulated by cauchy which describes the non-relativistic momentum transport in any continuum. 

50. De Moivre’s formula

The use of the process of mathematical induction can be used to prove a very significant theorem in mathematics called De Moivre’s theorem. If the complex number z = r(cos α + i sin α), then this pattern can be extended, using mathematical induction, to De Moivre’s theorem.

51. Fourier transform

The Fourier transform is a mathematical function that can be used to show the different parts of a continuous signal. It is used mostly to convert from time domain to frequency domain. Fourier transforms are frequently used to calculate the frequency range of a signal that changes over time. This kind of signal processing has many uses such as cryptography, oceanography, speech recognition, or handwriting recognition. Fourier transforms can also be used to solve differential equations. Fourier transform calculations requires understanding of integration and imaginary numbers. Computers are usually used to calculate Fourier transforms of complex signals. The Fast Fourier Transform is a method computers use to quickly calculate a Fourier transform.

52. Prime-counting function

In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by Ïe(x) (unrelated to the number Ie).

Number of primes in up to the number is x=x/lnx

53. MURPHY’S LAW FORMULA

Here, PM is the Murphy’s probability that something will go wrong. KM is Murphy’s constant (equal to one) and FM is Murphy’s factor, a very small number.

Murphy’s law states: “Anything that can go wrong will go wrong”.

54. SUMMATION FORMULA

In mathematics, summation (capital Greek sigma symbol: E) is the addition of a sequence of numbers; the result is their sum or total. If numbers are added in turn from left to right, any intermediate result is a partial sum. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. other types of values besides numbers can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid).

55. Logarithmic spiral A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which appears often in nature. The logarithmic spiral was first described by descarte and later deeply investigated by Jakob bernoulli, who called it  “the marvelous spiral”.

Logarithmic spirals in nature-

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons:

The approach of a hawk to its prey. The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. The arms of spiral galaxies.  The arms of tropical cyclones, such as hurricanes.  Many biological structures including spider webs and the shells of mullosks. 

56. Heron’s formula

Heron’s formula states that the area of a triangle whose sides have lengths a, b, and c is

A=s(s-a)(s-b)(s-c),

where a, b, c are the length of each of the triangle’s sides, and s is the semiperimeter of the triangle; that is,

s=(a+b+c)2.

In geometry, Heron’s formula gives the area of a triangle.

57. Quadratic equation

x=-b+/- sqrt(b^2-4ac)/2a

A quadratic equation is an equation in the form of ax2 + bx + c, where a is not equal to 0. It makes a parabola (a “U” shape) when graphed on a coordinate plane.

58. Euler line

In geometry, the Euler line, is a line found from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the 9-point circle of the triangle.

59. PYTHAGOREAN TRIPLES FORMULA

Euclid’s formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with m > n > 0. The formula states that the integers

a=m^2−n^2,

b=2mn,

c=m^2+n^2

form a Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

60. Euler’s formula

e^ix=cosx+isinx

Euler’s formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential functions. Euler’s formula is ubiquitous in mathematics, physics, and engineering. The physicist richard feynmann called the equation “our jewel” and “the most remarkable formula in mathematics”.

When

x=Ï

Euler’s formula evaluates to

e^i+1=0

which is known as ruler’s identity.

61.

Simplex method, Standard technique in linear programming for solving an optimization problem, usually one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region, and the solution is typically at one of the vertices. The simplex method is a systematic procedure for testing the vertices as possible solutions.

62.  PROOF INFINITE NUMBER OF PRIME NUMBERS

Theorem.

There are infinitely many primes.

Proof.

Suppose that p1=2 < p2 = 3 < … < pr are all of the primes. Let P = p1p2pr+1 and let p be a prime dividing P; then p can not be any of p1, p2, …, pr, otherwise p would divide the difference Pp1p2pr=1, which is impossible. So this prime p is still another prime, and p1, p2, …, pr would not be all of the primes.

63. Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:

Summation n=1 to infinity of 1/n=1+1/2+1/3+1/4+1/5+…

64. EULER SUMS

precise sum of the infinite series:

∑n=1 to ∞1/n^2=1/1^2+1/2^2+1/3^2+⋯=1.644934  or  π2/6

65. FORMULA FOR SOLUTION OF CUBIC EQUATION

In algebra, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d

in which a is nonzero.

Setting f(x) = 0 produces a cubic equation of the form

ax^3+bx^2+cx+d=0.

66. SOLUTION TO QUARTIC EQUATION

In algebra, a quartic function is a function of the form

f(x)=ax^4+bx^3+cx^2+dx+e,

where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.

The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals.

67. Quintic function

In algebra, a quintic function is a function of the form

g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,

where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other words, a quintic function is defined by a polynomials of degree five.

If a is zero but one of the coefficients b, c, d, or e is non-zero, the function is classified as either a quartic function, cubic function, quadratic function or linear function.

68. Lorentz force

In physics (particularly in electromagnetism) the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with velocity v in the presence of an electric field E and a magnetic field B experiences a force

F=qE+qv —B

(in SI units).

69. Euler-lagrange formula-

Lsubx(tsuby,q(t),qdot(t))-d/dtLsubx(t,q(t),qdot(t))=0

In the calculus of variation, the Euler-Lagrange equation, Euler’s equation, or Lagrange’s equation, is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. 

70. Euler product formula In number theory, an Euler product is an expansion of a dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive numbers raised to a certain powers proven by leonard euler. This series and its continuation to the entire complex plane would later become known as the riemann zeta function.

∏pP(p,s)

71. Euler-maclaurin formula In mathematics, the Euler-Maclaurin formula provides a powerful connection between integrals and sums. It is used to approximate integrals by finite sums, or  to evaluate finite sums and infinite series using integrals and  calculus. For example, many asymptotic expansions are derived from the formula, and faulhaber’s formula for the sum of powers is an immediate consequence.

72. Pi

pi=C/d

(pi is equal to the circumference divided by the diameter).

Pi is an endless string of numbers

Pi is a mathematical constant. It is the ratio of the distance around a circle to the circle’s diameter. This produces a number, and that number is always the same. This number starts 3.141592……. and continues without end, and are called irrational numbers.

The diameter is the longest straight line which can be fitted inside a circle. It passes through the center of the circle. The distance around a circle is known as the circumference. although the diameter and circumference are different for various circles, the number pi remains constant and its value never changes because the relationship between the circumference and diameter is always the same. the number pi was irrational; that is, it cannot be written as a fraction by normal standards, and it is part of the group of numbers known as transcendental, which are numbers that cannot be the solution to a polynomial equation. The properties of pi have allowed it to be used in many other areas of math besides geometry, which studies shapes. Some of these areas are complex analysis, trigonometry, and series. To find the area of a circle, use the formula  (radius²). This formula is sometimes written as A=r^2, where r is the variable for the radius of any circle and A is the variable for the area of that circle.

To calculate the circumference of a circle with an error of 1 mm:

4 digits are needed for a radius of 30 meters

10 digits for a radius equal to that of the earth

15 digits for a radius equal to the distance from the earth to the sun.

73. Exponential function

In mathematics, an exponential function is a function that quickly grows. More precisely, it is the function

exp(x)=e^x, where e is ruler’s constant, an irrational number that is approximately 2.71828.

74. Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. The natural logarithm of x is the power to which e would have to be raised to equal x.

75. Conic sections

CONIC SECTIONS-

Circle-

(x-g)^2+(y-h)^2=radius^2

(g=x coordinate, h=y coordinate)

Parabola-

y^2+/-4ax

(a=x coordinate)

x^2=+/-4ay

(a=y coordinate)

ellipse-

x^2/a^2+y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates)

Hyperbola-

x^2/a^2-y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates)

76. Exponential growth and decay

y=A*exp^k*t

A=starting number of for example bacteria, t=length of growth time, k=constant, y=number of bacteria after t time

77. Calculating an orbit, i.e. of a comet

Calculations: orbit, period of orbit, perihelion, aphelion and eccentricity

(for example a comet)—

Use ellipse formula x^2/a^2+y^2/b^2=1

Then calculate from 2 coordinates in AUs with formula

x^2 x b^2 + y^2 x a^2=a^2 x b^2

find a and b  (the closest and furthest approaches)

Period years of orbit^3=distance (a from above)^2

Period=cuberoot(distance AUs of â above)^2

Perihelion=d=a-c

c=(a^2-b^2)^1/2  (c=distance from focus to center of ellipse)

aphelion=d=c-b

perihelion=A x (1-eccentricity)

aphelion=A x (1+ eccentricity)

A (semimajor axis)=(perihelion + aphelion)/2

eccentricity=1-perihelion/A

eccentricity=aphelion/A-1

To find formula for the orbit, use ellipse formula

x^2/a^2+y^2/b^2=1, then use formula

x^2 x b^2+y^2 x a^2=a^2 x b^2,

Use 2 location coordinates from the orbit, plug in one of the coordinates

Into the 2nd formula, then plug in the 2nd coordinates into the same

formula. Subtract one of the resulting formulas from the other resulting

formula, then solve for a or b  with the formula that results from the

subtraction. Plug in the solution to a or b that was solved into one of the

Pre-subtraction formulas to find the a or b that has not been found yet.

Now, we have the a and b constants, so we plug them into the ellipse

Formula, and thus have the equation for the orbit of the stellar body,

I.e. a comet.

78. Interesting math example #1

1×1=1

11×11=121

111×111=12321

1111×1111=1234321

11111×11111=123454321

111111×111111=12345654321

Etc

79. Interesting math example #2

1×8+1=9

12×8+2=98

123×8+3=987

1234×8+4=9876

12345×8+5=98765

Etc

80. Interesting math example #3

1=.9999999

0.999

In mathematics, 0.999… (also written 0.9, among other ways), denotes the repeating decimal consisting of infinitely many 9 after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than all decimal number 0.9, 0.99, 0.999, etc.[1] This number can be shown to equal 1. In other words, “0.999…” and “1” represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proof. The technique used depends on target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999… is commonly defined. (In other systems, 0.999… can have the same meaning, a different definition, or be undefined.) More generally, every nonzero terminating decimals has two equal representations (for example, 7.52 and 7.51999..), a property true of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons such as rigorous proofs relying on non-elementary techniques, properties, and/or disciplines math students can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.

SCIENTIFIC FORMULAS—

MATHEMATICS

PHYSICS

ASTRONOMY

ROCKET SCIENCE

Capital Low-case Greek Name English
Alpha a
Beta b
Gamma g
Delta d
Epsilon e
Zeta z
Eta h
Theta th
Iota i
Kappa k
Lambda l
Mu m
Nu n
Xi x
Omicron o
Pi p
Rho r
Sigma s
Tau t
Upsilon u
Phi ph
Chi ch
Psi ps
Omega o

POWERS 

tera=10^12 

giga=10^9 

mega=10^6 

myria=10^4 

kilo=10^3 

hecto=10^2

icosa=20 

quindeca=15 

hendeca=11 

dec=10 

non=9 

octo=8 

hepta=7 

hexa=6 

penta=5 

tetra=4

tri=3

bi=2

uni=1

semi=.5

deci=10^-1

centi=10^-2

milli=10^-3

micro=10^-6

nano=10^-9

pico=10^-12

femto=10^-15

atto=10^-18

1 pound=.4545 kilograms

1 kilogram=2.2026432 pounds

1 mile=1.609 kilometers

1 kilometer=.62150404 kilometers

1 pound=16 ounces

1 pound=454.54 grams

1 ounce=28.040875 grams

Speed of light=299,792,458 meter/second

Temperature conversion—

From Fahrenheit to:

celsius=(Fahrenheit-32) x .5556

kelvin=(fahrenheit-32)/1.8+273.15

From celsius:

fahrenheit=(1.8 x celsius)+32

kelvin=celsius+273.15

From kelvin:

fahrenheit=1.8 x (kelvin-273.15)+32

celsius=kelvin-273.15

PRACTICAL MATHEMATICS FORMULAS PLATONIC SOLIDS—

1. Tetrahedron

Surface area=Sqrt3 x edge length^2 Volume=sqrt2/12 x edge length^3 2. Cube

Surface area=6 x edge length^2 volume=edge length^3

3. Octahedron

Surface area=2 x sqrt3 x edge length^2

volume=sqrt2/3 x edge length^3

4. Dodecahedron

Surface area=3 x sqrt(25+10 x sqrt5) x edge length^2 volume=(15+7 x sqrt5)/4 x edge length^3

5. Isocahedron

Surface area=5 x sqrt3 x edge length^2

volume=(5 x (3+sqrt5))/12 x edge

length^3

CIRCLE-

Diameter D = 2 x Radius

Circumference- C = 2 x Pi*Radius

area- A = Pi x Radius^2

SPHERE-

Surface area. A = 4 x Pi x Radius^2

volume V = 4/3 x Pi x Radius^3

Diameter of a sphere. d=cuberoot(3/4 x Pi x volume) x 2 SQUARE, RECTANGLE, PARALLELOGRAM

Area A=side 1 x side 2

VOLUME OF SQUARE, RECTANGLE, PARALLELOGRAM V=side 1 x side 2 x side

PYRAMID

Surface area=base area+.5 x slant length

Volume=base x depth x height/3

CYLINDER

Surface area=2 x pi x radius x (radius+height)

Volume=PI X radius^2 x length

CONE

Surface area=pi x radius x (radius+base to apex length) Volumes=Pi x radius^2 x height/3

TORUS

Surface area=4 x pi^2 x radius torus x radius of solid part volume=2 x pi^2 x radius torus x radius solid part^2 

PYTHAGOREAN THEORM-

a^2+b^2=c^2

a=length of one right angle’s leg

b=length of other right angle’s leg

c=length of hypotenuse

LAW OF SINES-

a/sinA=b/sinB=c/sinC=2 x R=a x b x c/2 x area of triangle R=(a x b x c)/(squareroot((a+b+c) x (a+b-c) x (b+c-a)) Area of triangle=1/2 x a x b x sinC

LAWS OF COSINE-

c^2=a^2+b^2-2 x a x b x cosC

cosC=(a^2+b^2+c^2)/2 x a x b

AREA OF A TRIANGLE-

area=base x height x 1/2

AREA OF AN EQUILATERAL TRIANGLE- area=(length of a side )^2 x SQRT(3)/4 AREA OF A TRAPEZOID-

A=(top side+bottom side) x height/2

HERON’S FORMULA (area of any triangle)- area=SRQT(s x (s-side 1) x (s-side 2) x (s-side 3)) s=1/2 x (a + b + c)

SLOPE-

m=(y-y1)/(x-x1)

(Y1 and x1 are locations on coordinate plane) POINT SLOPE EQUATION OF A LINE-

Y -y1=slope(x-x1)

(Y1 and x1 are locations on coordinate plane) SLOPE INTERCEPT FORM FOR A LINE- y=slope(x)+(y intercept)

DISTANCE FORMULA-

distance=square root((x-x1)^2+(y-y1)^2+(z-z1)^2)) (z1, y1, and x1 are locations on coordinate system) ALGEBRA FORMULAS-

(a+b)^2=a^2+2 x a x b+b^2

(a-b)^2=a^2-2 x a x b+b^2

x^2-a^2=(x+a) x (x-a)

x^3-a^3=(x-a) x (x^2+a x x+a^2) x^3+a^3=(x+a) x (x^2-a x x+a^2) 

a/b+c/d=(a x d+b x c)/b x d) a/b-c/d=(a x d-b x c)/b x d

a/b x c/d=a x c/b x d

CONIC SECTIONS-

Circle- (x-g)^2+(y-h)^2=radius^2

(g=x coordinate, h=y coordinate) Parabola-

y^2+/-4ax

(a=x coordinate)

x^2=+/-4ay

(a=y coordinate)

ellipse-

x^2/a^2+y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates) Hyperbola-

x^2/a^2-y^2/b^2=1,

(a=x, b=y coordinates, or a=y, b=x coordinates) QUADRATIC EQUATION- x=(-b+/-squareroot(b^2-4ac))/2a

e=1/ln

ln=1/e

exp=1/log

log=1/exp

LAWS OF EXPONENTS-

a^x x a^y=a^(x+y)

a^x/a^y=a^(x-y)

(a^x)^y=a^(X x Y) (a*b)^x=a^x x b^x a^0=1

a^1=a

LAWS OF LOGARITHMS-

log(base a)(M x N)=log(base a)(M)+log(base a(N) log(base a)(M/N)=log(base a)M-log(base a)(N) logM^r=r X x logM

log(base a)(M)=logM/loga

TRIGONOMETRY-

sine-o/h

cosine=a/h

tangent=o/a

cosecant=h/o

secant=h/a

cotangent=a/o

(a=adjacent side of right triangle)

(o=opposite side of right triangle)

(h=hypotenuse of right triangle)

Pythagorean identities-

sin^2(x)-cos^2(x)=1

sec^2(x)-tan^2(x)=1

csc^2-cos^2(x)=1

Product relations-

Sinx-tanx x cosx

cosx=cotx x sinx

tanx=sinx x secx

cotx=cosx x cscx

Secx-cscx x tanx

cscx=secx x cotx

Trigonometry functions-

sinx=x-x^3/3!+x^5/5!-x^7/7!

cosx=1-x^2/2!+x^4/4!-x^6/6!

Inverse trigonometry functions-

sin-1x=x+(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5+(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+… cos-1x=pi/2-(x+(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5+(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+… tan-1x=x-x^3/3+x^5/5-x^7/7+…

cot-1x=pi/2-x+x^3/3-x^5/5+x^7/7-…

Hyperbolic functions-

sinhx=x+x^3/3!+x^5/5!+x^7/7!+…

coshx=1+x^2/2!+x^4/4!+x^6/6!+…

Inverse hyperbolic functions-

sinh-1x=x-(1/2 x 3) x x^3+(1 x 3/2 x 4 x 5) x x^5-(1 x 3 x 5/2 x 4 x 6 x 7) x x^7+… tanh-1x=x+x^3/3+x^5/5+x^7/7+…

Nth TERM OF AN ARITHMETIC SEQUENCE- 

Nth term=a+(number of terms-1)*d

(a=1st term, d=common difference)

SUM OF n TERMS OF AN ARITHMETIC SERIES- 

Sum-n/2 x (a+nth term)

(a=1st term, d=common difference)

Nth TERMS OF A GEOMETRIC SEQUENCE-

a(n)=a x r^(n-1)

(r cannot equal 0.)

(a=1st term, r=common ratio, n=number of terms)

SUM OF THE n TERMS OF A GEOMETRIC SEQUENCE- 

sum=a x ((1-r)^n)/(1-r)

(r cannot equal 0, 1)

(a=1st term, n=number of terms, r=common ratio)

SUM OF AN INFINITE SERIES-

s=n/(1-r)

(If absolute value of r<1)

(n=number start with)

(r=how much keep multiplying by) (s=sum of infinite series)

COMBINATIONS-

C(n,r)=n!/r!(n-r)!

PERMUTATIONS-

P(n,r)=n!(n-r)!

BINOMIAL FORMULA-

(a x x-b)^n

CALCULUS (DIFFERENTIATION)-

d/dx (x^n)=n x x^(n-1)

d/dx sinx= cost

d/dx cosx= -sinx

d/dx tanx=sec^2(x)

d/dx cotx=-csc^2(x)

d/dx secx-secs x tanx

d/dx cscx= -cscx x cotx

d/dx e^x=e^x

d/dx lnx=1/x

d/dx (u+v)=du/dx+dv/dx

d/dx(c x u)=c x du/dx

(chain rule)

d/dx (u x v)=(v x (du/dx)-(u x (dv/dx)

(product rule)

d/dx(u/v)=(v x du/dx-u x dv/dx)v^2

(quotient rule)

du=du/dx(dx)

CALCULUS (INTEGRATION)-

The definite integral of t from a to b for definite integral f(t)=F(b)-F(a)

Indefinite integral of x^r dx=x^(r+1)/(r+1)+c, (r cannot equal -1)

Indefinite integral of 1/x dx=ln(absolute value (x))+c

Indefinite integral of sinx dx=cosx+c

Indefinite integral of cosx dx=sinx+c

Indefinite integral of e^x dx=e^x+c

Indefinite integral of (f(x)+g(x))dx=indefinite integral f(x)+indefinite integral g(x) Indefinite integral of c x f(x) dx=c x (indefinite integral f(x))

indefinite integral of (u)dv=u x v-indefinite integral (v)du

(integration by parts)

CENTER OF MASS-

Center of mass (x)=((mass1) x (center of mass1)+(mass2) x (center of mass x 2))/ (mass-1+mass-2)

(A point representing the mean position of the matter in a body of system.) 

Exponential growth/decay-

Final amount=starting quantity x e^(k x time)

(Growth if k>1, decay if k<1)

VECTOR ANALYSIS

Norm (magnitude of a vector)=sqrt(x^2+y^2+z^2)

Dot product u (dot) v=(u1) x (v1)+(u2) x (v2)+(u3) x (v3)=||u|| ||v|| cos(theta)

(theta is the angle between u and v, 0<=theta<=Pi)

Cross product u x v=((u2) x (v3)-(u3) x (v2))i-((u1) x (v3)-(u3) x (v1))j+((u1) x (v2)-

(u2) x (v1))k

||u x v||=||u|| x ||v|| sin(theta)

(Theta is angle between u and v, 0<=theta<=Pi)

2 vectors orthogonal if their dot product v and u=0 or transpose vector v and vector u=0.

Exponential growth and decay—

y=A exp^k*t

A=starting number of for example bacteria, t=length of growth time, k=constant, y=number of bacteria after t time

OUTLINE OF PHYSICS FORMULAS

*** Straight line motion- ***

Velocity (meters/second)=distance (meters)/time (seconds) v=d/t (constant velocity)

v=2 x d/t (accelerating)

Distance (meters)=velocity (meters/second) x time (seconds) d=v x t

time (seconds)=distance (meters)/velocity (meters/second) t=d/v

t=sqrt(2 x distance/acceleration)

Acceleration (meters/second^2)=

((meters/second (end)-meters/second (start)/)time (seconds))/2 a=(d2/t-d1/t)/2

Acceleration (meters/second^2)=2 x distance (meters)/time (second)^2 a=2 x d/t^2

Final velocity (meters/second)=initial velocity (meters/second)+ acceleration (meters/second^2 x time (seconds)

v(f)=v1+a x t

velocity^2=initial velocity+2 x acceleration x distance

v^2=v1+2 x a x d

Average velocity (meters/second)=initial velocity+final velocity/time v(average)=(v1+v2)/t

Average velocity=initial velocity+1/2 x acceleration x time v(average)=v1+1/2 x a x t

Distance (meters)=initial velocity x time+1/2 x acceleration x time^2 d=v1+1/2 x t x a x t^2

Distance=acceleration x time^2/2

d=a x t^2/2

Newton’s 2nd law of motion

Force (newtons)=mass (kilograms) x acceleration (meters/second^2) F-m x a

Falling bodies

velocity=gravity (9.81 meters/second^2 for the earth) x time

v=g x t

How far fallen in meters=1/2 x gravity x time^2 d=1/2 x g x t^2

Time fallen=sqrt(2 x height/gravity)

t=sqrt(2 x h/g)

velocity=sqrt(2 x gravity x height) v=sqrt(2 x g x h)

*** Circular motion- *** 

Uniform circular motion

Moment of inertia=mass x distance from axis^2

m(inertial)=m x d^2

Angular velocity=angular displacement/change in time (radians/second) v(angular)=d/t

Angular momentum=moment of inertia x angular velocity m(angular)=m(inertial) x v(angular)

Centripedal acceleration

Centripedal acceleration=velocity^2/radius of path (radians/second^2) a(centipedal)=v^2/r

Torque (newtons-meter)

Centripetal force

Centripetal force=mass x velocity^2/radius of path

f(centripedal)=m x v^2/r

Gravitation

gravitation (newtons)=G x (mass(1) x mass(2))/radius^2

(G=6.67 x 10^-11)

f=G x (m1 x m2)/r^2

Fundamental forces in nature— strong

W eak

Electromagnetic

Gravity

*** Energy- ***

work

work (joules)=force (newtons) x distance (meters)

w=f x d

work=work output/work input x 100%

w=w(o)/w(i) x 100

Power

Power (watts)=work (joules)/time (seconds)

p=w/t

horsepower=746 watts

weight=mass x gravity

w=m x g

momentum=mass (kilograms) x velocity (meters/second) momentum=m x v

Energy

kinetic energy

KE (joules)=1/2 x mass (kilograms)x velocity (meters/second)^2 ke=1/2 x m x v^2

Potential energy

PE (joules)=mass x gravity (9.81 meters/second^2) x height (meters)

pe=m x g x h

Rest energy

Rest energy (joules)=mass x 300,000,000^2

Conservation of energy

Momentum-Â (kilograms-meters/second)

Linear momentum

L. momentum=mass (kilograms)x velocity (meters/second) m(momentum)=m x v

Conservation laws

Conservation of mass-energy

Conservation of linear momentum

Angular momentum

Conservation of angular momentum

Conservation of electric charge

Conservation of color charge

Conservation of weal isospin

Conservation of probability

Conservation of rest mass

Conservation of baryon number

Conservation of lepton number

Conservation of flavor

Conservation of parity

Invariance of charge conjugation

Invariance under time reversal

CP symmetry

Inversion or reversal of space, time, and charge

(there is a one-to-one correspondence between each of the conservation laws and a differentiable symmetry in nature.)

Impulse

impulse=force (newtons) x time (seconds)

i=f x t

*** Relativity ***

special relativity

Lorentz transformation

General relativity

*** Fluids- ***

Density

Specific gravity

kilograms/meter^3

Pressure

pressure=force/area

pressure=newtons/meters^3

p=f/d^3

Pressure in a fluid

pressure=density (kilograms/meters^3) x depth (meters) x weight (kilograms) p=d(density) x d(depth) x w

p=kg/d^3 x d x m

Archimede’s principle-the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.
Bernoulli’s principles-an increase in the speed of a fluid occurs 

simultaneously with a decrease in pressure or a decrease in the fluid’s potential energy.
v^2/2+gz+p/Ï=constant

v is the fluid flow speed at a point on a streamline,

g is the gravitational acceleration

z is the elevation of the point above a reference plane,

with the positive z-direction pointing upward so in the direction

opposite to the gravitational acceleration,

p is the pressure at the chosen point, and

Ï is the density of the fluid at all points in the fluid.

*** Heat- ***

internal heat

Temperature

Heat

1 kilocalorie=3.97 british thermal units (BTU)

1 BTU=.252 kilocalories

Specific heat capacity

Heat transferred=mass (kilograms) x specific heat capacity x temperature change (kelvin)

h=m x h x t

change of state

Heat of fusion

Heat of vaporization

pressure and boiling point

*** Kinetic theory of matter- ***

Ideal gases

Boyle’s law

pressure(1) x volume(1)=pressure(2) x volume(2)

(temperature constant)

P1 x v1=p2 x v2

Absolute temperature scale

Temperature kelvin=temperature (celsius)+273.15

Charlie’s law

volume(1)/temperature(1)=volume(2)/temperature(2)

(pressure constant)

v1/t1=v2/t2

Ideal gas law

pressure(1) x volume(1)/temperature(1)=pressure(2) x volume(2)/temperature(2) P1 x v1/t1=p2 x v2/t2

Kinetic energy of gases

Molecular energy

KE (joules)=3/2 x K x temperature (kelvin)

(K=boltzmann’s constant=1.38 x 10^-23 joules/kelvin

ke=3/2 x k x t

solids and liquids

Atoms and molecules

*** Thermodynamics- ***

3 laws of thermodynamics

The four laws of thermodynamics are:

Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other.

This law helps define the notion of temperature.

1st law of thermodynamics: When energy passes, as work, as heat, or with matter, into or out from a system, the system’s inertial energy changes in accord with the law of conservation of energy. Equivalently, Perpetual motion machines of the 1st kind (machines that produce work without the input of energy) are impossible.

2nd law of thermodynamics: In a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases. Equivalently, perpetual motion machines of the 2nd kind (machines that spontaneously convert thermal energy into mechanical work) are impossible.

3rd law of thermodynamics: The entropy of a system approaches a constant value as the temperature approaches absolute zero. With the exception of non-crystalline solids (glasses) the entropy of a system at absolute zero is typically close to zero, and is equal to the logarithm of the product of the quantum ground states.

entropy

The entropy of a system approaches a constant value as the temperature

absolute zero.

Mechanical equivalent of heat

Mech. Equiv. heat=4,185 x joules/kilocalories Mech. Equiv. heat=778 x foot-pounds/BTU

Heat engines

Engine efficiency

efficiency=1-heat temperature absorbed/heat temperature given off eff=1-h(temp. Absorbed)/h(temp. Given off)

Conduction

Convection

Radiation

*** Electricity- ***

Electric charge

Charge of proton=1.6 x 10^-19 coulombs

Charge of electron= -1.6 x 10^-19 coulombs

Electric charge=current (amperes) x time taken (seconds)

Coulomb’s law

Electric force (newtons)=K x charge-1 (coulombs) x charge-2 (coulombs)/ distance (meters)^2

(K=9 x 10^9 newton-meter^2/coulomb^2)

F=KÂ x c1 x c2/d^2

Atomic structure

Mass of proton=1.673 x 10^-27 kilograms

Mass of neutron=1.675 x 10^-27 kilograms Mass of electron=9.1 x 10-31 kilograms Ions

Electric field

Electric field (newton/coulomb)=force (newtons)/charge (coulombs) E=f/c

force=charge x electric field

Electric lines of force

Potential difference

volts=work/charge

(1 volt= 1 joule/coulomb)

volt=electric field (newtons/coulomb)x distance (meters)

v=E x d

Electric field (newtons/coulombs)=volts/distance

E=v/d

Potential Difference=current (amperes)x resistance (ohms)=

energy transferred/charge (coulombs)

pd=I x r=e/c

Electric current-

Electrical energy=voltage (volts)x current (amperes)x time taken (seconds) e=v x I x t

Electric current

Electric current (amperes)=charge (coulombs)/time interval (seconds)

(1 ampere=1 coulomb/second)

I=c/t

Electrolysis

Ohm’s law

Electric current (amperes)=volts/resistance (ohms)

(resistance (1 ohm))=1 volt/ampere)

I=v/r

resistance (ohms)=voltage (volts)/current (amperes)

r=v/I

voltage (volts)=current (amperes)x resistance (ohms)

v=I x r

Resisters in series

resistance=resistance(1)+resistance(2)+resistance(3)

R=r1+r2+r3

Resisters in parallel 1/resistance=1/resistance(1)+r1/resistance(2)+1/resistance(3) 1/R=1/r1+1/r2+1/r3

Kirchoff’’s law

current law=Summation (current)=0

Voltage law=summation (voltage)=0

Capacitance

1 farad=1 coulomb has 1 volt between plates

capacitance (farad)=charge (coulombs)/voltage (volts)

Work stored=work (charging)=1/2 x capacitance x voltage^2

W=w=1/2 x C x v

electric power

power (watts)=work done per unit time (joules)=voltage (volts)x charge

(coulombs)/time (seconds)

p=w=v x c/t

Power (watts)=current (amperes) x voltage (volts)=current (amperes)^2

x resistance (ohms)=voltage (volts)^2/resistance (ohms)

p=i x pd=i^2 x r=pd^2/r

Alternating current

power (watts)=1/2 x peak voltage x peak current x

cos (phase angle between current and voltage sine waves)

p=1/2 x v x I x cos(theta)

*** Magnetism- ***

Magnetic field

1 tesla=1 newton/ampere-meter

(tesla=weber/meter^2)

(1 gauss=10^-4 teslas)

B (magnetic field)=K x I (straight line current)/distance (meters)

(K=2 x 10^-7 newton/amperes^2

B=K x I/d

Magnetic field on a moving charge

Magnetic field on a current

B=Pi(3.14) x K x I/r

F=I x L x B

Magnetic field of solenoid

B=2 x Pi x K x N (number of turns)/L (length of solenoid) x I (current amperes) Forces between 2 currents

(K=2 x 10^-7 newton/amperes^2

F/L=K x I(1) x I(2)/d

Lorentz force

F=charge x electric field+charge x velocity (cross product) magnetic field F=qE+qv x B

*** electromagnetism ***

Maxwell’s equations

Maxwell’s equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

*** Electromagnetic induction- *** 

Generator

Motors

Alternating current

I=I (max)/sqrt(2)=.707 I (max)

V=I (max)sqrt(2)=.707 V (max)

Transformer

Primary voltage/secondary voltage=primary turns/secondary turns Primary current/secondary current=secondary turns/primary turns DC circuits

*** Waves- ***

Frequency

1 hertz=1 cycle/second

W avelength

wave velocity (meters/second)=frequency (hertz) x wavelength (meters)

v=f x w

Acoustics

Optics

Electromagnetic waves

Velocity of light=c=3 x 10^8 meters/second=186,282 miles/second

Doppler effect

Frequency found by observer with respect to sound=frequency produced by source x (velocity sound+velocity observer)/(velocity sound-velocity source) f(o)=f(s) x (v+v(o))/(v-v(s))

Frequency found by observer with respect to light=frequency produced by source x sqrt((1+relative velocity/c)/(1-relative velocity/c))

f(o)=f(s) x sqrt((1+v/c)/(1-v/c))

reflection and refraction of light

Index refraction=n=c/velocity in medium (meters/second)

r=c/v

Interference, diffraction, polarization

Particles and waves

*** Quantum physics- ***

Uncertainty principle-the velocity of an object and its position

cannot both be measured exactly, at the same time, even in

theory.

The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wave function gives the probability of finding the particle at a certain position. … The solution to this equation is a wave that describes the quantum aspects of

a system.

The Pauli exclusion principle— 

Quantum theory of light

Quantum energy=E=h x f

Planck’s constant=h=6.63 x 10^-34 joules-second

X-rays

Electron volt=planck’s constant x frequency

eV=h x f

Electron KE=x-ray photon energy

f=eV/h

momentum=kinetic energy (joules)/speed of light^2 (meters/second) p=ke/c^2

Kinetic Energy=planck’s constant x frequency

planck’s constant x speed light/wavelength

ke (joules)=6.63 x 10^-34 joules-second x frequency hertz(meter/second)

The Pauli exclusion principle is the quantum mechanical

principle which states that two or more identical fermions (particles with

half-integer spin)

cannot occupy the same quantum state within a quantum system

simultaneously.

6.63 x 1-^-34 x 300,000,000 meters/f (meters) joules=h x f=h x c/lambda

Electron volt

1 eV=1.6 x 10^-19 joules

1 KeV=10^3 eV

1 MeV=10^6 eV

1 GeV=10^9 eV

Kinetic energy=1/2 x mass x velocity^2=

planck’s constant x frequency-electon volts

ke (joules)=1/2 x m x v^2=6.63 x 10^-34 joules-second x f-eV Matter waves

De Broglie wavelength=lambda=h/m x v (momentum=m x v)

wavelength=planck’s constant/(mass x meters/second) lambda=h/m x v

Solid state physics

***Nuclear and atomic physics- ***

Nucleus

Mass (proton)=1.673 x 10^-27 kilograms=1.007277 u

Mass (neutron)=1.675 x 10^-27 kilograms=1.008665 u Nuclear structure

Binding energy

Mass defect= change m=((number protons x mass hydrogen)+ (number neutrons x mass neutrons))-m

(mass hydrogen=1.007825 u)

Fundamental forces

Gravity

Electromagnetic

Weak interaction

Strong force

Fission and fusion

Radioactivity

Alpha particle=helium nuclei

Beta particle=electron

Gamma rays=high energy photons with frequencies greater than x-rays neutron>proton+electron

proton>neutron+positron Radioactive decay and half-life Elementary particles and antimatter

ASTRONOMY FORMULAS

1. How to find the DISTANCE in parsecs to a star- distance=10^((apparent magnitude-absolute magnitude+5)/5) 

2. APPARENT magnitude

apm=log d x 5-5+abm

apparent=log(distance) x 5 – 5 + absolute magnitude

3. How do you calculate the absolute magnitude of a star abm=((log L/log 2.516)-4.83)

Absolute magnitude=((log(number of sun’s luminosity of star)/ log2.516)-4.83

abm=-(5 x log d-5-apm)

Absolute magnitude= -(5 x log(distance parsecs)-5-apparent magnitude)

4. To find brightness of a star/number of suns-

L=10^((abm-4.83) x (log 2.516))

LUMINOSITY of star=10^-((absolute magnitude of star-4.83)*(log2.516))

luminosity increase=2.512^([magnitude increase]+4.83)

luminosity=mass^3.5Â (for main sequence stars)

Luminosity (watts)=4 x pi x radius(meters)^2 x temperature (kelvin)^4 x

5.67 x 10^-8 watts maters^-2 kelvin^-4

5. Mass binary system

Suppose in an example, we calculate the masses of 2 stars in a binary star system: if the period of star a is 27 years and its distance from the common center of mass is 19 AUs, the

Distance^3/period^2=19^3/27^2=6859/729=9.4 solar masses for the total mass of the 2 stars.

The velocity of star a is 30,000 km./second and star b is 10,000 km/second, so 30,000/10,000=3.

The mass of star b is 9.4/(1+3)=2.25 solar masses.

The mass of star a is 9.4-2.25=7.15 solar masses.

So star a is 7.15 solar masses, and star b is 2.25 solar masses, and both added up equals 9.4 solar masses, the combined mass of the 2 stars.

6. Radius of a star

radius=(temperature sun (kelvin)/temperature star (kelvin))^2 x (2.512^(absolute magnitude sun-absolute magnitude star)^1.2)

radius=sqrt(luminosity)/(temperature kelvin)^2

7. size of star/orbit/object-

size object miles=arcseconds size object x distance parsecs x 864,000

8. LT=10^10 x m(star)/m(sun)^-2.5

lifetime=(10^10) x (mass of star/mass of sun)^-2.5

9. To find ARC SECOND size—

Arcs second size=1/distance parsecs x number of suns size

9.5 DISTANCE (parsecs)– d=1/arcseconds

10. Galaxy distance in millions of light years-

d=13,680 x rsh+8.338

distance (millions of light years)=13,680 x red shift+8.336

10.5 size of an object I.e. galaxy=-number of megaparsecs x 1,000,000 x sin(arc seconds/360)

11. velocity of galaxy in kilometers/second-

v=300,000 x rsh

velocity (kilometers per second)=300,000 x redshift

12.  approximate number of stars in a galaxy=luminosity in number of suns galaxy/.02954

13. redshift-

Rsh=mly-8.338/13,680

redshift=(light years (millions)-8.336)/13,680

14. Volume of a galaxy=4/3 x Pi x a x b^2

a=major axis, b=minor axis, (for elliptical galaxies)

15. Number of stars=volume/distance between stars^3

16. Average Distance between stars in light years=cube root(volume/number of stars) 

16.5. Number of stars span across longest axis of galaxy=(3/4 x volume)/(Pi x a x b x n^2)

a=an axis length in light years, b=the other axis in light years, n=distance between stars light years

17. Escape velocity from a galaxy miles/second=

sqrt(.468 x 1.989 x 10^30 x 6.67 x 10^-11 x number of stars/(4 x pi/3)/8 x .71353 x 2.697823 x 10^20)/1612.9

18. Surface gravity

gravity(meters/second^2)=

mass of star number of suns x 1.99 x

10^30 x 6.67 x 10^-11/(size of star number of suns x 864000000 x 1.62/2)^2 

19. Titus-bode law-

Distance (astronomical units)=3*2^n+4/10

(n=-infinity, 0, 1, 2, 3, …)

Mercury=-infinity

Venus=0

earth=1

mars=2

Asteroid belt=3

jupiter=4

Etc

20. kepler’s 3 laws of planetary motion-

1. Planets travel in elliptical orbits.

2. Equal areas are covered in in equal times in the elliptical orbit.

3. The distance in astronomical units to the 3rd power equals the time to travel one complete orbit in years to the 2nd power.

(time years)^2=(radius orbit astronomical units)^3

D=P^(2/3)

P=D^(3/2)

21. Velocity to achieve orbit=sqrt(G x M/distance from center of the earth) 

22. Escape velocity=sqrt(G x M/r)

23. Four types of eccentric orbits

circle eccentricity=0

ellipse eccentricity= 0-1

parabola eccentricity=1

hyperbola eccentricity>1

24. Eccentricity=(greatest orbital distance-closest orbital distance)/(closest orbital distance+greatest orbital distance) e=(d(greatest)-d(closest))/(d(closest)+d(greatest))

25. Calculations: orbits, periods of orbits, perihelions, aphelions and eccentricities (for example a comet)

CIRCLE

Circular formula eccentricity=0

(x-h)^2+(y-k)^2=r^2. (#1)

(h=x coordinate and y=y coordinate ;r=radius of orbit)

period=2 x pi x sqrt(radius^3/(6.67 x 10^-11 x mass of body the body is orbiting)) 

Velocity in orbit=sqrt(6.67 x 10^-11 x

mass of body the body is orbiting/radius of orbit)

Centripetal acceleration=velocity^2/radius of orbit

ELLIPSE

Use ellipse formula x^2/a^2+y^2/b^2=1

Then calculate from 2 coordinates in AUs with formula

x^2 x b^2 + y^2 x a^2=a^2 x b^2

find a and b (the closest and furthest approaches)

Period years of orbit=distance (AU)^3/2

time=2 x pi x sqrt(a^3/G x M)

distance=period^2/3

velocity=sqrt(G x M x (2/r-1/a))

eccentricity=(0<e<1)

eccentricity=sqrt(1-b^2/a^2)

a (semimajor axis)=(perihelion + aphelion)/2

Semi minor axis (b)=sqrt(-(eccentricity^2-1) x a^2

Perihelion/aphelion=(1+eccentricity)/(1-eccentricity)=aphelion distance/perihelion distance

Perihelion distance=semimajor axis x (1-eccentricity)

aphelion distance=semimajor axis x (1+_eccentricity)

Simple way to calculate and orbit

1=X^2+Y^2. >Â 1=x^2+semimajor axis^2+y^2/semiminor axis^2

time in seconds to get to orbital position from 0 degrees going counterclockwise t=radians at current position of orbiting body/360 x sort(semi

major axis^3/6.67 x 10^-11 x 1.989 x 10^30)

to find the angle at which the orbiting body is at from 0 degrees going counterclockwise x^2=p^2+1

q=sqrt(p)

r=sqrt(x)

s=q/r

arcsin(s)=angle of the orbiting body with respect to the focal body

to find distance to orbiting body from foci

x^2=p^2+p^2

cosy=x^2/P^2 x x^2

m=arccosy

cosm=1/n

q=1/cosm

q=distance to orbiting body from foci

To find formula for the orbit, use ellipse formula

x^2/a^2+y^2/b^2=1

PARABOLA

eccentricity=1

y=a x X^2+b x X +c

Calculate from 2 coordinates of the body in its orbit (a and b coordinates in

both instances)

Solve for x, then y, and will have the formula for the parabolic orbit.

Then will have the equation of the orbit like the quadratic equation above with

numerical figures for a and b.

Velocity of body in parabolic orbit

v=sqrt(2 x 6.67 x 10^-11 x mass of body being orbited/radius of body)

Period of orbit does not have an orbit so undefined.

eccentricity=sqrt(1+b^2/a^2)

time in seconds to get to orbital position from 0 degrees going counterclockwise– t=radians at current position of orbiting body/360 x sqrt(semi

major axis^3/6.67 x 10^-11 x 1.989 x 10^30)

to find the angle at which the orbiting body is at from 0 degrees going counterclockwise– g^2+(distance between foci/2 (‘p))^2=d^2

(d^2-p^2-g^2)/(-2(p^2)(g^2))=cosx

cos-1x=angle x

p^2-1^2=g^2

(1^2-p^2-g^2)/(-2(p^2)(g^2))=cosL

cos-1L=angle L

90-x-L=m

m-x=n

n is the angle the orbiting body is at from 0 degrees going counterclockwise HYPERBOLA

eccentricity>1

x/a-y/b=1. (#1)

xb-ya=ab. (#2)

Calculate from 2 coordinates of the body in its orbit (a and b coordinates in both instances) using (equation #2).

Solve for x, then y, and will have the formula (for #1 above) for the hyperbolic orbit.

Then will have the equation of the orbit like equation #1 with numerical figures for a and b.

Velocity of body at infinity in a hyperbolic orbit

velocity at an infinite distances away=sqrt(velocity^2-escape-velocity^2)

Period of orbit does not have na orbit so undefined.

26. EXAMPLES OF CALCULATING ORBITS

STEPS TO CALCULATE AN ELLIPTICAL ORBIT

Suppose we measure 2 coordinates of a comet, one at (4AU,1AU) and the other at (0AU,3AU).

1) We plug in each of the coordinates, the 1st equals the x and the 2nd equals the y, each of the 2 coordinates into b^2 x x^2+a^2 x y^2=a^2 x b^2.

Then we subtract the 2 equations from each other. Next, we solve for both a and b. The resulting equation is x^2/2.376^2+y^2/1.68^2=1

and this is the equation of the orbit for the 2 given coordinates. The ellipse equation is x^2/2.376^2+y^2/1.68^2=1.

Solving for x and y gives

x=sqrt((1-5.65 x y^2)/2.8) and y=sqrt((1-2.8 x x^2/5.65).

2) perehelion=1.68AU, aphelion=2.376AU. (figures for constants a and b)

3) semimajor axis=A=(perihelion+aphelion)/2=(1.68+2.376)/2=1.9958AU 

4) Semi minor axis=smaller figure of a and b=1.68AU.

5) focii=amaller figure of a and b=1.68AU.

6) distance center of ellipse to the foci=A-perihelion=.3158AU

7) period=aphelion^3/2=2.82 years (1,029.83 days)

8) eccentricity=1-perihelion/A=.158

9) velocity at any instant=1.99 x 10^30 x 6.67 x 10^-11 x (1/radius=1/A)=

At perihelion=20.07 miles/second, at aphelion=10.74 miles/second

10) Suppose we want to find the time from perihelion to a distance of 2 AU and its velocity there. We use the formulas

Semimajor axis^2+semimajor axis^2=x^2

cosx=x^2/2 x semimajor axis^2 x x^2; angle=arccosx

time=angle/360 x sqrt(semimajor axis^3/6.67 x 10^-11 x 1.989 x 10^30)

The answer for time at 2 AU=82.79/360 x sqrt(2.7186 x 10^34/1.32733 x 10^20)=

.22997 x 14311435.6=38.09 days at a velocity there at 12.995 miles/second.

HALLEY’S COMET

period=75.986 years.

focii=.6AU.

perihelion=.6AU.

aphelion=35.28AU.

Semimajor axis=17.94Au

eccentricity=.9855.

Velocity at perihelion=33.23 miles/second, aphelion=3.124 miles/second

STEPS TO CALCULATE A CIRCULAR ORBIT

Suppose 2 coordinates were recorded of a celestial body, one (3,2,646), and the other (1,3.873). It the celestial body has a circular orbit, the squares of each set of coordinates added together will equal the same defendant number. In this example,

3^2+2.646^2=15, and 1^2+3.873^2=16, so this is a circular orbit where the equation of the orbit is 4^2=x^2+y^2, where the 4 in the 4^2 is the radius of the circle, and the x and y are the

coordinates of the circular orbit. The eccentricity of of a circular orbit is equal to zero. STEPS TO CALCULATE A PARABOLIC ORBIT

Formulas-

velocity=sqrt(2 x G x Mass central body/radius of orbiting body from central body) trajectory=(4.5 x G x M x Time seconds^2)^1/3

To find out whether 2 coordinates measurements of an orbiting body if a parabolic orbit, say coordinates (3,27) and (2,12), we need to set up a parabolic equation

y=b x x^2, then put into it separately the 2 coordinates. 27=b x 3^2, solve for b to arrive at equals to 3.

12=b x 2^2, b also equals 3.

Since b in both equations are equal to each other, the orbiting object is in a parabolic orbit. The equation for the orbit is y=3 x x^2. The period of orbit is infinitely long since the orbiting object never returns. The vertex of the orbit is (0,0). The foci is equal to 3/4. We arrive at this by always using 4 x p, and setting it equal to 3, the number equal to b. When 4 x p=3 is solved, p=3/4. So the focus is at (0,3/4).the velocity of the orbiting body at its closest approach to the central body is equal to 30.05 miles/second.

STEPS TO CALCULATE A HYPERBOLIC ORBIT

Suppose 2 coordinates of an orbiting celestial body are recored as being at (28.28, 10)AU and (34.64, 14.14)AU positions. We try using the hyperbolic equation

1=x^2/a^2-y^2/b^2, solve for a and b, and the result equals 1, so this is a hyperbolic orbit. a=20 and b=10. The equation for the orbit is

1=x^2/20^2-y^2/10^2.

Solving for x and y yields

x=sqrt(400-4 x y^2), and y=sqrt(x^2/4-100(.

The orbit’s eccentricity is equal to sqrt(a^2+b^2)/a=sqrt(400+100)/400=1.118, which is greater than 1, so this is a hyperbolic orbit.

For focii=sqrt(a^2+b^2)=22.33AU from the sun’s position, which is equal to 22.33- a=22.33-20=20AU, which makes the focci=(20,0).

The period of this orbiting body is undefined since it will never return. To find the semimajor axis, we use the velocity formula

v=sqrt(6.67 x 10^-11 x 1.99 x 10^30 x (2/r-1/semimajor axis)).

r=2.33AU in meters and v=618,000 meters/second. Solving for the semimajor axis yields -1208.12.

Let us determine the velocity of the orbiting object at say 5AU from the sun.

v=sqrt(6.67 x 10^-11 x 1,99 x 10^30 x (2/5AU in meters-1/1208.12))=18.77 kilometers/second, or 11.64 miles/second.

If we wanted to calculate the velocity of the object when it gets as far away as the nearest star, 4.3 light years away,

it would be traveling 25.76 meter/second there.

27. Formulas to find masses, radius, and luminosities of WHITE DWARFS 

(mass<=.75 suns)

(radius<=.00436 suns)

(luminosity<=.00365 suns)

radius=mass^18.68 mass=radius^.052966

luminosity=mass^19.5198 mass=luminosity^.05123

luminosity=radius^1.04494 radius=luminosity^.95699

28. Formulas to find masses, radius, and luminosities of MAIN SEQUENCE stars luminosity=mass^3.5

mass=luminosity^(.2857)

radius=(temperature kelvin sun/temperature kelvin star)^2 x

(2.512^(absolute magnitude sun-absolute magnitude star)

note main sequence star’s masses can be found by knowing the star’s luminosity and its temperature.

Type star  Mass radius Temperature  luminosity  lifespan

O. +16. +6.6 +33,000 kelvin 55,000 to >200,000 >9.77 m/yrs

B 2.1-16. 1.8-6.6 10,000-33,000 kelvin. 42-24,000. 9.77 m/yrs-1.57 b/yrs

A 1.4-2.1. 1.4-1.8. 7,500-10,000 kelvin. 5.1-24. 1.57 b/yrs-4.3 b/yrs F. 1.04-1.4. 1.15-1.4. 6,000-7,500 kelvin. 2.4-5.1. 4.3 b/yrs-9.07 b/yrs

G. .8-1.04. .96-1.15. 5,200-6,000 kelvin .38-1.2. 9.07 b/yrs-17.47 b/yrs K. .45-.8. .7-.96. 3,700-5,200 kelvin. .08-.38. 17.47 b/yrs-73.62 b/yrs M. <=.45. <=.7. 2,000-3,700 kelvin <.002-.08. >73.62 b/yrs

29. Formulas to find SUBGIANT masses, radii, and luminosities 

mass^.94475=radius

radius^1.0585^mass 

mass^.84=radius

mass^4.2585=luminosity

luminosity^.2348=mass

radius^4.5076=luminosity

luminosity^.22185=radius

30. Masses, radius, and luminosities for GIANT stars- 

mass^1.7226=radius

radius^.58052=mass

mass^2.429=luminosity

luminosity^.4117=mass

radius^1.41=luminosity

luminosity^.7092=radius

31. Masses, radius, and luminosities for BRIGHT GIANT stars- 

mass^1.8578=radius

radius^.5383=mass

mass^5.2365=luminosity

luminosity^.19097=mass

radius^2.8186=luminosity

luminosity^.35397=radius

32. Finding masses, radius, and luminosities for SUPERGIANT stars—

mass^1.836=radius

radius^.547=mass

mass^3.84=luminosity

luminosity^.26=mass

radius^.209=luminosity

luminosity^.4785=radius

34. MASSES OF STARS AND THEIR FATES 

Masses .07-10 suns white dwarf

Masses .5-8 suns planetary nebulas

Masses >8 suns supernovas

Masses 10-29 suns neutron stars

masses>29 suns black holes

35.  Spectral type, temperature, color, mass, size, luminosity, % of stars

O 30,000 K blue >16 >6.6 >30,000 .00003%

B 10,000-30,00 blue white 2.1-16 1.8-6.6 25-30,000 .13%

A 7,500-10,000 white blue 1.4-2.1 1.4-1.8 5-25 .6%

F  6,000-7,500 yellow white 1.04-1.4 1.15-1.4 1.5-5 3%

G  5,200-6,000 yellow .8-1.04 .96-1.15 .6-1.5 7.6%

K 3,700-5,200 light orange .45-.8 .7-.96 .08-.6 12.1%

M 2,400-3,700 orange red .08-.45 <=.7 <=.08 76.45%

The Hertzsprung Russell diagram relates stellar classification with absolute magnitude, luminosity, and surface temperature.

36. DISTRIBUTION OF TYPES OF STARS IN GALAXY- 

Giants and supergiants .946%

O star .0000256% 

B stars .1105%

A stars .51085%

F stars 2.5545%

G stars 6.446%

K stars 10.313%

M stars 65.0295%

White dwarfs 8.515%

Brown dwarfs 1-10% (4.98% average estimate)

Neutron stars .8515%

Black holes .08515

DRAKE EQUATION ESTIMATE OF PERCENTAGE OF ADVANCED CIVILIZATIONS OF SYSTEMS STARS

Applies to 10% of all stars

36.5 averages:luminosities, masses, radius’s of stars in galaxies—

Average luminosity\=.271 suns

Average mass=.468 suns

Average radii=<.71353 suns

37. SEVERAL ABSOLUTE AND APPARENT MAGNITUDES WITH LUMINOSITIES LIST ABSOLUTE MAGNITUDES LIST

Gamma ray burst. -39.1 374,000 trillion suns

Quasars -33. 1,360 trillion suns

supernovas. -19.3. 4.49 billion suns

Supernova 1978a. -15.66. 157 million suns

Pistol star. -10.75 1.7 million suns

deneb -8.38. 192,424 suns

Betelgeuse. -5.5. 13,558 suns

Sun 4.83. 1 sun

Proxima centauri. 11.13. 1/331 suns

Sun in Andromeda galaxy. 29.07. 1/4.98 billion suns

Venus 29.23. 1/5.8 billion suns

Hubble telescope viewing limit. 31. 1/29.4 billion suns

James webb telescope viewing limit. 34. 1/466 billion suns

APPARENT MAGNITUDES LIST

sun. -26.72. 23.74 trillion suns

Full moon. -12.6. 9.38 million suns

Venus. -4.4. 4,922 suns

Sirius. -1.6. 373 suns

Most energetic gamma ray burst 12.2 billion light years away- 3.77. (374,000 trillion suns)

Sun seen by us if it were in andromedas galaxy. 53.31 1/(2.48 x 10^19) suns

Type 2 supernova in Andromeda as seen from here—  apparent magnitude—  4.94

Venus in Andromeda as seen from here—  apparent magnitude—  53.47

Deneb in Andromeda as seen from here—  apparent magnitude—  15.86

38. Formulas to find temperature kelvin from spectral class—

Temp.=1500 x spectral class number+10000

O0=20, O1=19,…, B9=1, A0=0

Temp.=187.27 x spectral class number+5880

A0=22, A1=21,…, G1=1, G2=0

Temp.=132.22 x spectral class number+3500

G2=18, G3=17,…, K9=1, M0=0, M1=-1, M2=-2,…, M9=-9

39. Weight in tons per teaspoon of a neutron star, black hole, etc—

Weight/teaspoon=(1.989 x 10^30 x number of sun’s masses/907.185)/

                             (4 x pi/3 x (radius of star x 693979234.4)^3 x 202884.202)

                                                          Spaceflight formulas


Meaning of variables in the formulas— 

v=velocity (meters/second)
vi=velocity initial (meters/second) 

vf=velocity final (meters/second) 

vexh=exhaust velocity (meters/second) 

isp=seconds
m=mass (kilograms)
mi=initial mass (kilograms)
mf=final mass (kilograms)
mr=mass ratio
a=acceleration (meters/second^2) 

f=force (newtons)
d=distance (meters) 

t=time (seconds)
ke=kinetic energy (joules)
ed=energy density (joules/kilograms) 

p=power (watts)
spp=specific power (kilowatts/kilograms) 

mm=momentum (meters x kilograms) 

i=impulse (thrust x seconds)
fr=fuel rate (kilograms/second) 

mw=molecular weight 

texh=temperature exhaust (kelvin) 

eff=propulsive efficiency
r=radius (meters)
ecc=eccentricity
g=acceleration due to gravity
(9.81 meters/second^2) 

At 1 meter/second velocity—

1 newton=2.2 pounds of thrust

1 newton=62.3689 ounces of thrust

1 newton=997.37 grams of thrust

.45 newtons=1 pound of thrust

Rocket equation
velocity
v=vexh x ln(mi/mf)
v=(d x 2)/t (when accellerating) 

v=d/t (constant velocity) 

v=sqrt(2 x a x d) 

Velocity of exhaust 

vexh=v/ln*(mi/mf) 

vexh=.25 x sqrt(texh/mw) 

Vexh (km/sec)=.0806 x sqrt(temperature kelvin)

Isp
isp=vexh/9.81
isp=f/(fr x 9.81)
isp=vf/(ln(mr) x 9.81)

isp=vf/ln(mr) x 9.81

isp=f x t

isp=f x t
Mass ratio

mr=e^(vf/isp x 9.81)
mr=mi/mf
mr=e^(v/vexh)
mr=e^(vf/(isp x 9.81))
Mass final
mf=mi/(e^(vf/vexh)
Mass initial
mi=mf x e^(vf/vexh)
mass
m=f/a
m=2 x ke/v^2
Force
f=m*a
f=ke/d
f=9.81 x isp x fr
acceleration
a=f/m
Energy
ke=1/2 x v^2 x m
ke=d x f

Energy density (rest mass energy) 

ed=ke/m 

Fuel flow rate 

fr=f/vexh 

fr=m/t 

Distance 

d=v x t/2 (with respect to accelerating) 

d=v x t (constant velocity) 

d=ke/f 

d=v^2/2 x a 

Time

t=(d x 2)/v (constant acceleration) 

t=d/v (constant velocity) 

t=((m x vf)^2/2)-(m x mi)^2/2) x (1/f) x (2/vi+vf) 

Power p=ke/t 

p=f x d/t 

Specific power 

sp=(p/1000)/m 

sp=f/mi

Momentum 

M=v x m

Impulse â 

i=f x t

Antimatter needed (kilograms) 

m=ke/1.8 x 10^16 

Propulsive efficiency 

eff=2/(1+(vexh/vf)) 

eff=(vf/vexh)^2/(e^(vf/vexh)-1)
(Maximum efficiency for ratio- vf/vexh<1.6) 

eff=f x g x isp/2 x p 

                                      UNIVERSAL PHYSICAL CONSTANTS

ATOMIC MASS UNITS- 

                     1.6605402 x 10^-27 kilograms

                     (1/12 0f the mass of an atom of carbon-12)

AVOGADRO NUMBER- 

                      6.0221367 x 10^23/moles

                      (mole=number of elementary entities that are in 

                      carbon-12 atoms in exactly 12 grams of carbon-12)

BOHR’S MAGNETON-

                      9.2740154 x 10^-24 joules/tesla

                      (The magnetic moment of an electron caused by either its orbital 

                      or spin angular momentum. Magnetic moment is a quantity that 

                      determines the torque it will experience in an external magnetic 

                      field. Torque is rotational force. A joule is equal to the work done on  

                      an object when a force of 1 newton acts on the object in the direction 

                      of motion through a distance of 1 meter: kilograms x meters^2/

                      seconds^2. A joule is also equal to 10 million ergs.  A Tesla is a 

                      derived unit of the strength of a magnetic field: kilograms/(seconds^2 

                      x amperes.)

BOHR RADIUS- 

                      5.29177249 x 10^-11 meters

                      (The mean radius of an electron around the nucleus of a hydrogen atom 

                      at its ground state.)

BOLTZMANN CONSTANT- 

                      1.3806513 x 10^-23 joules/kelvin

                      (A physical constant relating the average kinetic energy of 

                      particles in a gas with the temperature of the gas. Kelvin 

                      temperature scale is the primary unit of temperature measurement 

                      in the physical sciences, but is often used in conjunction with the 

                      celsius degree, which is of the same magnitude absolute zero in 

                      kelvin is equal to -273.15 degrees celsius.)

ELECTRON CHARGE-

                      1.60217733 x 10^-19 coulombs

                      (Charge carried by a single electron. The coulomb is the quantity of 

                      charge that has passed through the cross section of an electrical 

                      conductor carrying one ampere within one second.)

ELECTRON CHARGE/MASS RATIO- 

                      1.75881962 x 10^11 coulombs/kilograms

                      (The importance of the charge-to-mass ratio, according to classical 

                      electrodynamics, is that 2 particles with the same charge-to-mass ratio 

                      move in the same path in a vacuum when subjected to the same electric 

                      and magnetic fields.)

ELECTRON COMPTON WAVELENGTH-

                      2.42631058 X 10^-12 meters 

                      (A compton wavelength of a particle  is equal to the wavelength of a 

                      photon whose energy is the same as the mass of the particle. The 

                      compton wavelength of an electron is the characteristic length scale of 

                      quantum electrodynamics. It is the length scale at which relativistic 

                      quantum field theory becomes crucial for its accurate description.)

ELECTRON MAGNETIC MOMENT- 

                      9.2847701 x 10^-24 joules/tesla

                      (The electron is a charged particle of -1e, where e is the unit of 

                      elementary charge. Its angular momentum comes from 2 types of 

                      rotation: spin and orbital motion.)

ELECTRON MAGNETIC MOMENT IN BOHR MAGNETONS- 

                      1.00159652193

                      (Bohr magneton is a physical constant and natural unit for expressing the 

                      magnetic moment of an electron caused by either its orbital or spin 

                      angular momentum. The electron magnetic moment, which is the 

                      electron’s intrinsic spin magnetic moment, is approximately one Bohr 

                      magneton.)

ELECTRON MAGNETIC MOMENT/PROTON MAGNETIC MOMENT- 

                       658.21068801

ELECTRON REST MASS-  

                       9.1093897 x 10^-31 kilograms

ELECTRON REST MASS/PROTON REST MASS-  

                       5.44617013 x1 0^-4

                       This is how much less mass the electron is as compared to the proton. 

                       (1,836.21 times lighter than proton)

FARADAY CONSTANT-

                        9.6458309 x 10^4 coulombs/mole

                        (The magnitude of electric charge per mole of electrons.)

FINE STRUCTURE CONSTANT-

                          .00729735308

                           (The strength of the electromagnetic interaction between elementary 

                           particles.)

                        

                  

GAS CONSTANT-

                           8.3144710 x 10^joules/(mole x kelvin)

                           (A physical constant which is featured in many fundamental equations 

                           in the physical sciences, such as the ideal gas law and the Nernst 

                           equation.)

GRAVITATIONAL CONSTANT-  

                          6.67206 x 10^-11 newtons x meters^3/(kilograms*second^2) 

                          Denoted by letter G, it is an empirical physical constant involved 

                          in the calculation of gravitational effects.

IMPEDENCE IN VACUUM-  

                          3.767303134 x 10^2 ohms

                          (The wave-impedence of a plane wave in free space. Electric field 

                          strength divided by the magnetic field strength.)

                      

SPEED OF LIGHT-  

                          299,792,458 meters/second

SPEED OF LIGHT IN A VACUUM SQUARED-  

                        89,875,517,873,681,764 meters^2/seconds^2

MAGNETIC FLUX QUANTUM- 

                        2.06783383 x 10^-15 webers

                        (The measure of the strength of a magnetic field over a given area

                        taken perpendicular to the direction of the magnetic field.)

MOLAR IDEAL GAS VOLUME-

                        22.41410×10^-3 meters^3/moles

                        (As all gases that are behaving ideally have the same number density, 

                        they will all have the same molar volume. It is useful if you want to 

                        envision the distance between molecules in different samples.)

MUON REST MASS-

                        1.8835327×10^-28 kilograms

                        (A muon is an elementary particle similar to an electron, with an electric 

                        charge of -1 and a spin of 1/2, but with a much greater mass.)

NEUTRON COMPTON WAVELENGTH-

                         1.31959110 x 10^-15 meters

                         (Explains the scattering of photons by electrons. The compton 

                         wavelength of a particle is equal to the wavelength of a photon 

                         whose energy is the same as the mass of the particle.)

NEUTRINO REST MASS-  

                        3.036463233*10^-35 kilograms

NUCLEAR MAGNETON-

                        5.0507866 X 10^-27 Henry/meters

                        (A physical constant of magnetic moment. Using the mass of a proton, 

                        rather than the electron, used to calculate the Bohr magneton. unit of 

                        magnetic moment, used to measure proton spin and approximately 

                        equal to 1.1,836 Bohr magneton.)

                      

PERMEABILITY CONSTANT-

                        12.5663706144 x 10^-7 Henry/meters

                        (Magnetic constant, or the permeability of free space, is a measure of

                        the amount of resistance encountered when forming a magnetic field 

                        in a classical vacuum.)

PERMITTIVITY CONSTANT-

                        8.854187817 x 10^-12 farad/meters

                        (A constant of proportionality that exists between electric displacement 

                        and electric field intensity in a given medium.)

      

PLANCK’S CONSTANT-

                         6.6260755×10^-34 joules/hertz 

                         6.62607004×10^-34 meters^2 x kilograms/seconds

                         (This constant links the about of energy a photon carries with the 

                         frequency of its electromagnetic wave.)

PROTON COMPTON WAVELENGTH-

                        2.4263102367 x 10^-12 meters

                        (The compton wavelength is a quantum mechanical property of a 

                        particle. A convenient unit of length that is characteristic of an elementary            

                        particle, equal to Planck’s constant divided by the product of the particles 

                        mass and the speed of light.)

PROTON MAGNETIC MOMENT- 

                        1.41060761 x 10^-26 joules/tesla

                        (The dipole of the proton. Protons and neutrons, both nucleons, 

                        comprise the nucleus of an atom, and both nucleons act as small 

                        magnets whose strength is measured by their magnetic moments.)

PROTON MAGNETIC MOMENT IN BOHR MAGNETONS-

                        1.521032202 x 10^-3

                        (A physical constant and the natural unit for expressing the magnetic 

                        moment of an electron caused by either its orbit or spin angular 

                        momentum. 

PROTON MASS/ELECTRON MASS-

                        1,836.152701

PROTON REST MASS-  

                        1.6726231 x 10^-27 kilograms

RYDBERG CONSTANT-  

                        1.0973731534 x 10^7/meters

                        (A physical constant relating to atomic spectra, in the science of 

                        spectroscopy. Appears in the Balmer formula for spectral lines of the 

                        hydrogen atom.)

RYDBERG ENERGY-

                        13.6056981 electron-volts

                        (It corresponds to the energy of the photon whose wavenumber is the 

                        Rydberg constant, I.e. the ionization of the hydrogen atom. It describe

                        the wavelengths of spectral lines of many elements.)

STEFAN-BOLTZMANN CONSTANT-

                        5.67051 x 10^-8 weber/(meters^2 x kelvin^4)

                        (The power per unit area is directly proportional to the 4th power of the 

                        thermodynamic temperature.  It is the total intensity radiated over all 

                        wavelengths as the temperature increases, of a black body which is 

                        proportional to to 4th power of the thermodynamic temperature.)

Table of physical constants 

Universal constants 

Value 

Quantity Symbol 

299 792 458 m⋅s−1defined newtonian gravitational constant G 

Relative standard uncertainty 

Speed of light in a vacuum

6.67408(31)×10−11 m3⋅kg−1⋅s−2 Planck’s constant  

6.626 070 040(81) × 10−34 J⋅s. Reduced planck’s constant=h/2 x pi 

1.054 571 800(13) × 10−34 J⋅s. 

4.7 × 10−5 1.2 × 10−8 

1.2 × 10−8 

Electromagnetic constants Quantity 

Symbol 

Value (SI units)
Relative standard uncertainty

Magnetic constant (vacuum permeability)μ0
4π × 10−7 N⋅A−2 = 1.256 637 061… × 10−6 N⋅A−2defined

Electric constant (vacuum permittivity)ε0=1/μ0c2 8.854 187 817… × 10−12 F⋅m−1defined

Characteristic impedance of vacuum Z0=μ0c
376.730 313 461… Ωdefined

Coulomb’s constant ke=1/4πε0 

8.987 551 787 368 176 4 × 109 kg⋅m3⋅s−4⋅A−2defined

Elementary charge e 

1.602 176 6208(98) × 10−19 C.

Bohr magneton μB=eħ/2me 9.274 009 994(57) × 10−24 J⋅T−1 

conductance quantuum

7.748 091 7310(18) × 10−5 S. 

inverse conductance quantum 

G0−1=h/2e2 

12 906.403 7278(29) Ω 2.3 × 10−10

Josephson constant

kJ=2e/h 

4.835 978 525(30) × 1014 Hz⋅V−1 6.1 × 10−9
magnetic flux quantum 

φ0=h/2e 

2.067 833 831(13) × 10−15 Wb 6.1 × 10−9

nuclear magneton

μN=eħ/2mp 

5.050 783 699(31) × 10−27 J⋅T−1 6.2 × 10−9 

von Klitzing constant

RK=h/e2

25 812.807 4555(59) Ω 2.3 × 10−10 

6.1 × 10−9 

6.2 × 10−9 2.3 × 10−10 

Atomic and nuclear constants Quantity 

Symbol 

Value (SI units)
Relative standard uncertainty bohr radius

a0=α/4πR∞ 

5.291 772 1067(12) × 10−11 m 2.3 × 10−9

classical electron radius

re=e2/4πε0mec2m_ 

2.817 940 3227(19) × 10−15 m 6.8 × 10−10

electron mass 

me 

9.109 383 56(11) × 10−31 kg 1.2 × 10−8
fermi coupling constant

GF/(ħc)3

1.166 3787(6) × 10−5 GeV−2 5.1 × 10−7
fine-structure constant

α=μ0e2c/2h=e2/4πε0ħc 

7.297 352 5664(17) × 10−3 2.3 × 10−10
Hartree energy 

Eh=2R∞hc 

4.359 744 650(54) × 10−18 J 1.2 × 10−8
proton mass

mp 

1.672 621 898(21) × 10−27 kg 1.2 × 10−8
quantum of circulation 

h/2me 

3.636 947 5486(17) × 10−4 m2 s−1 4.5 × 10−10
Rydberg constant

R∞=α2mec/2h 

10 973 731.568 508(65) m−1 5.9 × 10−12
Thomson cross section

(8π/3)re2 

6.652 458 7158(91) × 10−29 m2 1.4 × 10−9
weak mixing angle  

sin2θW=1−(mW/mZ)2 0.2223(21)

9.5 × 10−3 

efimov factor

22.7 

Physico-chemical constants Quantity 

Symbol Relative standard uncertainty 

Value (SI units)

Atomic mass constant

mu=1u 

1.660539040(20)×10−27 kg 1.2×10−8
avagadro’s constant 

NA,L 

6.022140857(74)×1023 mol−1 1.2×10−8 

boltzmann’s constant

k=kB=R/NA 

1.38064852(79)×10−23 J⋅K−1 5.7×10−7
faraday’s constant 

F=NAe 

96485.33289(59) C⋅mol−1 6.2×10−9
first radiation constant 

c1=2πhc2 

3.741 771 790(46) × 10−16 W⋅m2 1.2 × 10−8 

for spectral radiance 

c1L=c1/π 

1.191 042 953(15) × 10−16 W⋅m2⋅sr−1 1.2 × 10−8
loschmidt constant
atT = 273.15 K and p = 101.325 kPa

n0=NA/Vm 

2.686 7811(15) × 1025 m−3 5.7 × 10−7
gas constant

R

8.3144598(48) J⋅mol−1⋅K−1 5.7×10−7 

molar Planck constant 

NAh 

3.990 312 7110(18) × 10−10 J⋅s⋅mol−1 4.5 × 10−10
molar volume of an ideal gas
atT = 273.15 K and p = 100 kPa 

Vm=RT/p 

2.271 0947(13) × 10−2 m3⋅mol−1
5.7 × 10−7
at T= 273.15 K and p= 101.325 kPa2.241 3962(13) × 10−2 m3⋅mol−1 5.7 × 10−7
sacker-tetrode constant
at 

T= 1 K and p= 100 kPa S0/R=52/R=+ln[(2πmukT/h2)3/2kT/p] 

−1.151 7084(14)1.2 × 10−6at T= 1 K and p = 101.325 kPa −1.164 8714(14)1.2 × 10−6
second radiation constant 

c2=hc/k 

1.438 777 36(83) × 10−2 m⋅K 5.7 × 10−7
stefan-boltzmann constant

σ=π2k4/60ħ3c2 

5.670367(13)×10−8 W⋅m−2⋅K−4 2.3×10−6 

wien displacement law constant 

b energy=hck−1/=hck^ 4.965 114 231…

2.8977729(17)×10−3 m⋅K
5.7×10−7
Wien’s entropy displacement law constant

b entropy=hck−1/=hck^ 4.791 267 357… 

3.002 9152(05) × 10−3 m⋅K 5.7 × 10−7
Adopted values Quantity 

Symbol
Value (
SI units)
Relative standard uncertainty
conventional value of josephson constant

KJ−90 

4.835 979 × 1014 Hz⋅V−1
0 (defined)
conventional value of von klitzing constant

RK−90 

25 812.807 Ω 

0 (defined)

constant

Mu=M(12C)/12 

1 × 10−3 kg⋅mol−1 0 (defined)
of carbon-12

M(12C)=NAm(12C) 

1.2 × 10−2 kg⋅mol−1 0 (defined)

Molar mass

standard acceleration of gravity (gee, free-fall on Earth) gn 

9.806 65 m⋅s−20 (defined) standard atmosphere

atm 

101 325 Pa 0 (defined) 

ATOMIC MASS UNITS-
1.6605402 x 10^-27 kilograms 

(1/12 0f the mass of an atom of carbon-12) 

AVOGADRO NUMBER-
6.0221367 x 10^23/moles 

(mole=number of elementary entities that are in carbon-12 atoms in exactly 12 grams of carbon-12) 

BOHR’S MAGNETON-
9.2740154 x 10^-24 joules/tesla 

(The magnetic moment of an electron caused by either its orbital
or spin angular momentum. Magnetic moment is a quantity that determines the torque it will experience in an external magnetic
field. Torque is rotational force. A joule is equal to the work done on an object when a force of 1 newton acts on the object in the direction of motion through a distance of 1 meter: kilograms x meters^2/ seconds^2. A joule is also equal to 10 million ergs. A Tesla is a derived unit of the strength of a magnetic field: kilograms/(seconds^2 x amperes.) 

BOHR RADIUS-
5.29177249 x 10^-11 meters 

(The mean radius of an electron around the nucleus of a hydrogen atom at its ground state.) 

BOLTZMANN CONSTANT-
1.3806513 x 10^-23 joules/kelvin 

(A physical constant relating the average kinetic energy of particles in a gas with the temperature of the gas. Kelvin temperature scale is the primary unit of temperature measurement in the physical sciences, but is often used in conjunction with the celsius degree, which is of the same magnitude absolute zero in kelvin is equal to -273.15 degrees celsius.) 

COSMOLOGICAL CONSTANT- 

R=1/2Rg=8 x pi x 6.67 x 10^-11 x T 

(T=energy-momentum tensor)
The constant is a homogeneous energy density that causes the expansion of the universe to accelerate. 

ELECTRON CHARGE-
1.60217733 x 10^-19 coulombs 

(Charge carried by a single electron. The coulomb is the quantity of charge that has passed through the cross section of an electrical conductor carrying one ampere within one second.) 

ELECTRON CHARGE/MASS RATIO-
1.75881962 x 10^11 coulombs/kilograms 

(The importance of the charge-to-mass ratio, according to classical electrodynamics, is that 2 particles with the same charge-to-mass ratio move in the same path in a vacuum when subjected to the same electric and magnetic fields.) 

ELECTRON COMPTON WAVELENGTH- 2.42631058 X 10^-12 meters 

(A compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the mass of the particle. The compton wavelength of an electron is the characteristic length scale of quantum electrodynamics. It is the length scale at which relativistic quantum field theory becomes crucial for its accurate description.) 

ELECTRON MAGNETIC MOMENT- 9.2847701 x 10^-24 joules/tesla 

(The electron is a charged particle of -1e, where e is the unit of elementary charge. Its angular momentum comes from 2 types of rotation: spin and orbital motion.) 

ELECTRON MAGNETIC MOMENT IN BOHR MAGNETONS- 1.00159652193 

(Bohr magneton is a physical constant and natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum. The electron magnetic moment, which is the electron’s intrinsic spin magnetic moment, is approximately one Bohr magneton.) 

ELECTRON MAGNETIC MOMENT/PROTON MAGNETIC MOMENT- 658.21068801 

ELECTRON REST MASS-
9.1093897 x 10^-31 kilograms 

ELECTRON REST MASS/PROTON REST MASS- 5.44617013 x1 0^-4 

This is how much less mass the electron is as compared to the proton. (1,836.21 times lighter than proton) 

FARADAY CONSTANT-
9.6458309 x 10^4 coulombs/mole 

(The magnitude of electric charge per mole of electrons.) 

FINE STRUCTURE CONSTANT- .00729735308 

(The strength of the electromagnetic interaction between elementary particles.) 

GAS CONSTANT-
8.3144710 x 10^joules/(mole x kelvin) 

(A physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation.) 

GRAVITATIONAL CONSTANT-
6.67206 x 10^-11 newtons x meters^3/(kilograms*second^2) 

Denoted by letter G, it is an empirical physical constant involved in the calculation of gravitational effects. 

IMPEDENCE IN VACUUM-
3.767303134 x 10^2 ohms 

(The wave-impedence of a plane wave in free space. Electric field strength divided by the magnetic field strength.) 

SPEED OF LIGHT-
299,792,458 meters/second 

SPEED OF LIGHT IN A VACUUM SQUARED- 89,875,517,873,681,764 meters^2/seconds^2 

MAGNETIC FLUX QUANTUM-
2.06783383 x 10^-15 webers 

(The measure of the strength of a magnetic field over a given area taken perpendicular to the direction of the magnetic field.) 

MOLAR IDEAL GAS VOLUME- 22.41410×10^-3 meters^3/moles 

(As all gases that are behaving ideally have the same number density, they will all have the same molar volume. It is useful if you want to envision the distance between molecules in different samples.) 

MOLAR MASS CONSTANT- 1×10^-3 kilograms/moles (relates relative atomic mass and molar mass) 

MOLAR MASS OF CARBON-12— 1.2×10^-2 kilograms/moles (relates atomic mass of carbon-12 and molar mass) 

MUON REST MASS-
1.8835327×10^-28 kilograms 

(A muon is an elementary particle similar to an electron, with an electric charge of -1 and a spin of 1/2, but with a much greater mass.) 

NEUTRON COMPTON WAVELENGTH- 1.31959110 x 10^-15 meters 

(Explains the scattering of photons by electrons. The compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as the mass of the particle.) 

NEUTRINO REST MASS- 3.036463233*10^-35 kilograms 

NUCLEAR MAGNETON-
5.0507866 X 10^-27 Henry/meters 

(A physical constant of magnetic moment. Using the mass of a proton, rather than the electron, used to calculate the Bohr magneton. unit of magnetic moment, used to measure proton spin and approximately equal to 1.1,836 Bohr magneton.) 

PERMEABILITY CONSTANT-
12.5663706144 x 10^-7 Henry/meters 

(Magnetic constant, or the permeability of free space, is a measure of the amount of resistance encountered when forming a magnetic field in a classical vacuum.) 

PLANCK CHARGE- 1.875545956×10^-18 coulombs (a quantity of electric charge) 

PERMITTIVITY CONSTANT-
8.854187817 x 10^-12 farad/meters 

(A constant of proportionality that exists between electric displacement and electric field intensity in a given medium.) 

PLANCK’S CONSTANT-
6.6260755×10^-34 joules/hertz 

6.62607004×10^-34 meters^2 x kilograms/seconds
(This constant links the about of energy a photon carries with the frequency of its electromagnetic wave.) 

PLANCK CONSTANT (REDUCED)- 6.582119514×10^-16 eV-seconds 

(h-bar, in which h equals h divided by 2pi, is the quantization of angular momentum.) 

PLANCK’S LENGTH- 1.616229X10&-35 meters
(the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate.) 

PLANCK MASS- 2.17647X10^-8 kilograms
(derived approximately by setting it as the mass whose compton wavelength and schwarzschild radius are equal.) 

PLANCK TIME- 5.3916×10^-44 seconds
(time needed for light to travel 1 planck length in a vacuum.) 

PLANCK TEMPERATURE 1.416808×10^32 degrees kelvin
(if an object were to reach this temperature, the radiation it would emit would have a wavelength of 1.616×10^-35 meters, Planck’s length, at which point quantum gravitational effects become relevant.) 

PROTON COMPTON WAVELENGTH- 2.4263102367 x 10^-12 meters 

(The compton wavelength is a quantum mechanical property of a particle. A convenient unit of length that is characteristic of an elementary particle, equal to Planck’s constant divided by the product of the particles mass and the speed of light.) 

PROTON MAGNETIC MOMENT-
1.41060761 x 10^-26 joules/tesla 

(The dipole of the proton. Protons and neutrons, both nucleons, comprise the nucleus of an atom, and both nucleons act as small magnets whose strength is measured by their magnetic moments.) 

PROTON MAGNETIC MOMENT IN BOHR MAGNETONS- 1.521032202 x 10^-3 

(A physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbit or spin angular momentum. 

RYDBERG CONSTANT-
1.0973731534 x 10^7/meters 

(A physical constant relating to atomic spectra, in the science of spectroscopy. Appears in the Balmer formula for spectral lines of the hydrogen atom.) 

RYDBERG ENERGY-
13.6056981 electron-volts 

(It corresponds to the energy of the photon whose wavenumber is the Rydberg constant, I.e. the ionization of the hydrogen atom. It describe the wavelengths of spectral lines of many elements.) 

STANDARD ACCELERATION ON EARTH BY GRAVITY- 9.80665 meters/seconds^2 

STANDARD ATMOSPHERE- 101.325 pascals 

(pressure, temperature, density, and viscosity of the earth’s atmosphere.) 

STEFAN-BOLTZMANN CONSTANT-
5.67051 x 10^-8 weber/(meters^2 x kelvin^4) 

(The power per unit area is directly proportional to the 4th power of the thermodynamic temperature. It is the total intensity radiated over all wavelengths as the temperature increases, of a black body which is proportional to to 4th power of the thermodynamic temperature. This constant is used to link a star’s temperature to the amount of light it emits.) 

MAGNETIC CONSTANT (VACUUM PERMIABILITY)- 1.256637061X10^-6 NEWTONS/AMPERES^2 

ELECTRIC CONSTANT (VACUUM PERMITTIVITY)- 8.854187817X10^-12 FARAD/METER 

CHARACTERISTIC IMPEDANCE OF VACUUM- 376.730313461 OHMS 

COULOMB’S CONSTANT-
8.9875517873881764X10^8 KILOGRAMS METERS^3/ 

SECONDS^4XAMPERES^2 

ELEMENTARY CHARGE-
1.6021766208X10^-19 COULOMBS 

CONDUCTIVE QUANTUM-
7.748091731X10^-8 SECONDS 

INVERSE CONDUCTIVE QUANTUUM- 12.9064037278 OHMS 

JOSEPHSON CONSTANT-
4.835978525X10^14 HERTZ/VOLTS 

MAGNETIC FLUX QUANTUM- 2.067831X10^-15 WEBERS 

NUCLEAR MAGNETON-
5.050783699X10^-27 JOULES/TESLAS 

VON KILTZING CONSTANT- 258.12807557 ohms 

CLASSICAL ELECTRON RADIUS- 2.8179403227X10^-15 METERS 

ELECTRON MASS-
9.10938356×10^-31 kilograms 

FERMI COUPLING CONSTANT- 1.1663787X10^-5 GeV^-2 

FINE-STRUCTURE CONSTANT- 7.2972525664X10^-3 

HARTREE ENERGY-
4.35974465X10^-18 JOULES 

PROTON MASS- 

1.6726219×10^-27 kilograms 

QUANTUM OF CIRCULATION-
3.6369475486X10^-4 METERS^2/SECONDS 

THOMSON CROSS SECTION- 6.6524587158X10^-29 METERS^2 

WEAK MIXING ANGLE- .2223 

EFIMOV FACTOR- 22.7 

FIRST RADIATION CONSTANT-
3.74177179X10^-16 WEBER-METERS^2 

FIRST RADIATION CONSTANT (for spectral radiance)- 1.191042953×10^-16 webers-meters^2-seconds/radius 

Loschmidt constant-
2.6867811×10^25 /meters^3 

Molar planck constant-
3.990312711×10^-10 joules-seconds/moles 

Molar volume of an ideal gas (at t=273.15 k and p=100 kpa)- 2.2710947×10^-2meters^3/moles 

Molar volume of an ideal gas (at t=273.15 k and p=101.325 kpa)- 2.2413962×10^-2 meters^3/moles 

Sacker-tetrode constant (at t=273.15 k and p=100 kpa)- -1.1517084 

Sacker-tetrode constant (at t=273.15 k and p=101.325 kpa)- -1.1648714 

Second radiation constant-
1.438777×10^-2 meters kelvin 

Wien displacement law constant- 2.8977729×10^-3 meters kelvin 

Wien entropy displacement constant- 3.0029152×10^-3 meters kelvin 

Conventional value of Josephson constant- 4.835979×10^14 hertz/volts 

Conventional value of von Klitzing constant- 25812.807 ohms 

2 PARAMETERS OF THE HIGGS FIELD POTENTIAL- V(H)=lambda*(R^2-v^2)=lambda*H^4-2*v^2*H^2*lambda*v^4
(H is the higgs field)
the higgs field is an energy field that is thought to exist everywhere in the universe. the field is accompanied by a fundamental particle called the higgs boson, which the field uses to continuously interact with other particles. the process of giving a particle mass is known as the higgs effect. 

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