astronomy formulae

Astronomical formulae

Formulas compiled by Frank Sorensen 2002-2003. Last changed 2004-10-19. All page- or equation numbers refer to Carroll & Ostlie: "An introduction to modern astronomy", Addison-Wesley Publishing Company Inc. 1996 Equations without numbers or references are from my own notes.

Side 1 af 36

Constants

Radiation constant =

The speed of light =

Gravitational constant = Plancks constant =

Boltzmann's constant =

Luminosity of the sun =

Epoch correction = Mass of electron =

Mass of Hydrogen atom =

Mass of proton =

Mass of sun = Absolute bolometric magnitude of the sun =

Epoch correction = Period of the Earth =

Universal gas constant = Rydberg constant =

Stefan-Boltsman constant =

Cross section for Thompson electron scattering =

Side 2 af 36

Variables

acceleration

Area

Average distance from the sun

Semimajor axis in ellipse

Semimajor axis of object 1 (known)

Semimajor axis of object 2 (unknown)

Centripetal acceleration

Interstellar extinction coefficient

Semimajor axis of reduced mass

Angle subtended by semimajor axis

Right ascension

Angle subtended by semimajor axis 1 (known)

Angle subtended by semimajor axis 2 (unknown)

Apparent blue magnitude

Semiminor axis in ellipse

Blackbody intensity at wavelength lamda at Temperature T

Bolometric correction

Specific heat at constant pressure

Specific heat at constant volume

Ionization energy

Aperture

Distance [cm]

Distance [AU]

Distance [pc]

Absorptionline spreading

Differential vertical optical depth

Differential optical depth

Declination

Full width of 21 cm line at half maximum

Change in right ascension

Proper motion

Change in declination

Change in wave length

Time interval

Differential intensity

Side 3 af 36

Differential Luminosity

Differential mass interval

Differential mass interval inside star

Differential pressure interval Luminosity gradient

Differential radiation pressure

Differential heat interval

Differential radius interval

Differential distance

Differential entropy interval

Differential temperature interval

Differential time interval

Differential speed interval

Differential volume interval

Differential hydrogen mass fraction

Eccentricity

Total mechanical energy of system

Energy of a photon

Energy generation rate

Energy generation rate due to gravity

Focal ratio

Force

Radient Flux

Flux of object 1 (known)

Flux of object 2 (unknown)

Focal length of eyepiece

Focal length

Focal length of objective

Radiation pressure

Flux at surface

Acceleration of gravity on the surface of star/planet

Ratio of specific heats

Adiabatic limit

Pressure scale height

Projection angle

Initial intensity at wave length lambda

Intensity at wavelength lambda

Emission coefficient

Kinetic energy

Side 4 af 36

Opacity

Opacity at wavelength lambda

Luminosity

Steplength

Luminosity of object 1 (known)

Luminosity of object 2 (unknown)

Maximum radiative luminosity

Approximate ratio of proton potential and kinetic energy

Wavelength

Wavelength of maximum intensity

Observed wavelength

Wavelength at rest

Angular magnification

Bolometric absolute magnitude

Bolometric apparent magnitude

Main quantum state number

Total mass of star/planet/atom/object

Bolometric absolute magnitude of object 1 (known)

Bolometric magnitude of object 1 (known)

Mass of object 1 (known)

Bolometric absolute magnitude of object 2 (unknown)

Bolometric magnitude of object 2 (unknown)

Mass of object 2 (unknown)

Average mass of gas particle

Absolute blue magnitude

Jeans mass

Mass inside radius

Absolute ultraviolet magnitude

Absolute visual magnitude

Mean molecular weight

Proper motion

Reduced mass


Number of atoms pr cm^3

Number of illuminated lines in the grating

Number of particles/atoms

Number of steps

Order of diffraction lines

Side 5 af 36

Time difference in years (Epoch)

Total number density pr volume

Column density of neutral hydrogen (in unit 1/cm)

Number of atoms in ionization state i

Index of refraction at wavelength lambda

Number of particles moving at the speed v

Frequency

Momentum

Paralax in arcseconds

Period

Pressure

Pressure of free electrons

Radiation pressure

Nuclear energy generated pr particle

Orbit radius

Radius of/inside star/planet

Distance to center of mass of object 1 (known)

Radius of curvature of surface 1 (known)

Distance to center of mass of object 2 (unknown)

Radius of curvature of surface 2 (unknown)

Distance at aphelion

Jeans radius

Distance at perihelion

Density

Distance traveled

Synodic period

Cross section for bound-free photoionization

Collision cross section

Temperature / Effective temperature (Kelvin)

Free-fall time for a homologous collapse

Optical depth of 21 cm line center

Optical depth

Incident angle

Smallest angular seperation

Apparent ultraviolet magnitude

Inclination of rotational axis

Internal energy

Potential energy

Side 6 af 36

Apparent visual magnitude

Orbital velocity

Speed

Speed of stellar wind

Speed of star 1 (known) in binary system

Speed of star 2 (unknown) in binary system

Velocity at aphelion

Rotational speed at equator

Escape velocity

Most probable speed

Velocity at perihelion

Radial velocity of object 1 (known)

Radial velocity of object 2 (unknown)

Radial velocity

Root mean-squared speed of particles

Tangential velocity

Hydrogen Mass fraction

Helium mass fraction

Metal mass fraction

Redshift

Partition function

Side 7 af 36

Formulas

acceleration

Eq. p0034 Newtons 2nd law

Area

Eq. p0074 (3.13) Radiation pressure (absorption)

Eq. p0073 (3.14) Radiation pressure (reflection)

Eq. p0077 (3.16) Stefan-Boltsman equation

Average distance from the sun

Eq. p0052 (2.35) a Keplers 3rd law

Eq. p0050 (2.30) Orbital velocity at perihelion

Eq. p0051 (2.31) Orbital velocity at aphelion

Semimajor axis in ellipse

Eq. p0028 Phythagorean theorem

Eq. p0030 (2.5) Distance at perihelion

Eq. p0030 (2.6) Distance at aphelion

Eq. p0206 Angle subtended by semimajor axis

Semimajor axis of object 1 (known)

Eq. p0206 a Angle subtended by semimajor axis 1

Eq. p0206 c Semimajor axis of reduced mass

Semimajor axis of object 2 (unknown)

Eq. p0206 b Angle subtended by semimajor axis 2


Side 8 af 36

Centripetal acceleration

Eq. p0039 b Radial acceleration

Interstellar extinction coefficient

Eq. p0438 (12.1) Effect of interstellar extinction

Semimajor axis of reduced mass


Angle subtended by semimajor axis


Eq. p0206 (7.2) Mass ratio of double stars

Right ascension

Eq. p0016 (1.2) Change in right ascension by epoch

Eq. p0016 (1.3) Change in declination by epoch

Angle subtended by semimajor axis 1 (known)



Angle subtended by semimajor axis 2 (unknown)


Apparent blue magnitude

Eq. p0083 a U-B Color index

Semiminor axis in ellipse


Side 9 af 36

Blackbody intensity at wavelength lamda at Temperature T

Eq. p0081 (3.20) The Planck function

Bolometric correction

Eq. p0083 b Bolometric correction

Specific heat at constant pressure

Eq. p0354 (10.72) Ratio of specific heats

Specific heat at constant volume


Ionization energy

Eq. p0234 (8.7) Saha

Aperture

Eq. p0166 (6.6) Reyleigh criterion

Eq. p0173 (6.7) Focal ratio

Distance [cm]

Eq. p0277 (9.23) Random walk displacement

Distance [AU]

Eq. p0064 a Distance in AU from parallax in arc seconds

Distance [pc]

Eq. p0019 (1.4) Proper motion

Side 10 af 36

Eq. p0064 (3.1) Distance from paralax in arcseconds

Eq. p0067 (3.2) Radient Flux

Eq. p0068 (3.6) Distance modulus




Eq. p0438 (12.1) Effect of interstellar extinction

Absorptionline spreading

Eq. Additional 1 Absorptionline spread due to rotation

Differential vertical optical depth

Eq. p0286 (9.36) The radiation field

Differential optical depth

Eq. p0266 (9.13) Optical depth

Declination


Full width of 21 cm line at half maximum

Eq. p0445 (12.4) Optical depth of 21 cm line center

Change in right ascension


Proper motion

Eq. p0018 Proper motion

Side 11 af 36

Change in declination


Change in wave length

Eq. p0110 (4.39) b Redshift

Eq. p0126 (5.1) a Doppler shift

Eq. p0130 (5.2) Grating resolution

Time interval


Differential intensity

Eq. p0280 (9.26) Pure emission Eq. p0280 (9.27) Emission and absorption

Differential Luminosity

Eq. p0342 (10.45) Luminosity gradient

Eq. p0341 a Energy generation rate

Differential mass interval

Eq. p0319 (10.8) Mass conservation equation


Eq. p0409 (11.4) Mass loss rate

Differential mass interval inside star

Eq. p0384 a Mass conservation (10.8)

Differential pressure interval Luminosity gradient

Eq. p0318 (10.6) Radial acceleration

Side 12 af 36

Eq. p0318 (10.7)

Hydrostatic equilibrium Constraints:

Hydrostatic equilibrium

Eq. p0352 (10.26) Pressure scale height

Eq. p0360 (10.87) Condition for convection

Differential radiation pressure


Differential heat interval

Eq. p0366 (10.95) Energy generation rate due to gravity

Differential radius interval


Eq. p0318 (10.7)




Eq. p0351 Temperature gradient




Eq. p0356 (10.81) Adiabatic temperature gradient


Differential distance


Side 13 af 36



Differential entropy interval


Differential temperature interval




Differential time interval



Eq. Additional 2 Hydrogen depletionrate


Differential speed interval

Eq. p0225 (8.1)

Maxwell-Boltzmann distribution function

Differential volume interval


Differential hydrogen mass fraction


Side 14 af 36

Eccentricity






Total mechanical energy of system

Eq. p0056 (2.46) Total mechanical energy of system

Energy of a photon

Eq. p0131 (5.3) Energy of a photon


Energy generation rate




Energy generation rate due to gravity


Focal ratio


Force


Radient Flux

Side 15 af 36




Flux of object 1 (known)

Eq. p0067 (3.3) Flux ratio


Flux of object 2 (unknown)



Focal length of eyepiece

Eq. p0174 (6.9) Angular magnification

Focal length

Eq. p0161 (6.2) Lensmakers formula


Focal length of objective


Radiation pressure



Side 16 af 36

Eq. p0287 (9.37) Radiative flux


Flux at surface

Eq. p0077 (3.18) Surface Flux of a star

Acceleration of gravity on the surface of star/planet

Eq. p0039 (2.12) Acceleration of gravity

Eq. p0318 (10.7) Hydrostatic equilibrium Constraints:


Eq. p0352 (10.63)

Pressure scale height Constraints:


Ratio of specific heats




Adiabatic limit


Pressure scale height


Eq. p0352 (10.63)



Projection angle

Eq. p0211 (7.8)

The mass function of a binary system m1 comparable to m2

Side 17 af 36

Eq. p0210 (7.7) Sum of masses in a binary system

Eq. p0210 (7.8) a

Mass function assuming m2 << m1 Constraints:

m2 << m1 - m2 is negliable compared to m1

Initial intensity at wave length lambda

Eq. p0267 (9.16)

Intensity of ray travelling through gass from an optical depth

Intensity at wavelength lambda

Eq. p0267 (9.16)


Eq. p0280 (9.27) Emission and absorption

Emission coefficient


Kinetic energy

Eq. p0056 (2.45) The virial theorem

Opacity



Eq. p0463 (12.19) Maximum radiative luminosity

Opacity at wavelength lambda

Eq. p0266 a Mean free path for photons



Luminosity

Side 18 af 36


Eq. p0068 (3.8) Absolute bolometric magnitude of a star


Eq. p0077 (3.17) Luminosity of a spherical star


Steplength



Eq. p0266 b Mean free path for photons

Luminosity of object 1 (known)

Eq. p0068 (3.7) Ratio of luminosities

Luminosity of object 2 (unknown)


Maximum radiative luminosity


Approximate ratio of proton potential and kinetic energy

Eq. p0411 a Approximate ratio of proton potential and kinetic

energy at radius r0 Eq. p0411 (11.8) Pressure structure of corona

Wavelength


Eq. p0081 Frequency

Side 19 af 36

a



Eq. p0131 c Frequency - wave length

Eq. p0135 (5.7) Balmer lines

Eq. p0135 (5.8) a Lyman lines

Eq. p0135 (5.8) b Paschen lines

Eq. p0135 (5.8)

Generalized energylevels in the hydrogen atom


Eq. p0270 Cross section for bound-free ionization of hydrogen in the n'th quantum state

Wavelength of maximum intensity

Eq. p0076 (3.15) Wiens displacement law

Observed wavelength


Wavelength at rest




Angular magnification


Side 20 af 36

Bolometric absolute magnitude


Eq. p0068 (3.8) Absolute bolometric magnitude of a

star

Eq. p0083 b Bolometric correction Eq. p0438 (12.1) Effect of interstellar extinction

Bolometric apparent magnitude


Eq. p0083 b Bolometric correction Eq. p0438 (12.1) Effect of interstellar extinction


Eq. p0135 (5.8) Generalized energylevels in the hydrogen atom

Total mass of star/planet/atom/object



Eq. p0043 (2.17) Escape velocity



Eq. p0225 (8.1)


Eq. p0229 (8.2) Most probable speed of a particle

Eq. p0229 (8.3)

Root-mean-squared speed of thermal particles

Eq. Notes 1 Number of particles


Side 21 af 36

Bolometric absolute magnitude of object 1 (known)


Bolometric magnitude of object 1 (known)



Mass of object 1 (known)


Eq. p0046 Reduced mass


Eq. p0210 Mass ratio of binary stars

Eq. p0052 (2.35) b

Keplers 3rd law assuming circular orbit Constraints:

Circular orbit - Radius is constant

Eq. p0211 (7.8)


Eq. p0210 (7.5) Ratio of radial velocities


Eq. p0210 (7.8) a



Bolometric absolute magnitude of object 2 (unknown)


Bolometric magnitude of object 2 (unknown)


Side 22 af 36


Mass of object 2 (unknown)





Eq. p0052 (2.35) b



Eq. p0211 (7.8)




Eq. p0210 (7.8) a



Average mass of gas particle

Eq. p0323 (10.13) Mean molecular weight

Absolute blue magnitude


Jeans mass

Eq. p0449 (12.7) Jeans mass

Mass inside radius


Side 23 af 36

Eq. p0318 (10.7)




Absolute ultraviolet magnitude


Absolute visual magnitude


Mean molecular weight

Eq. p0323 (10.13) Mean molecular weight

Eq. p0323 (10.14)

Ideal gass law Constraints:

Ideal gas - The ideal gaslaw applies

Eq. p0325 (10.19) Mean molecular weight for neutral gas

Eq. p0326 (10.21)

Mean molecular weight for completely ionized gas

Eq. p0328 (10.26)

Total pressure (gas + radiation) Constraints:




Eq. p0353 (10.65)

Internal energy Constraints:

Nonreletavistic gass


Eq. p0449 (12.8) Jeans radius

Proper motion

Side 24 af 36


Reduced mass



Eq. p0135 (5.7) Balmer lines

Eq. p0135 (5.8) a Lyman lines

Eq. p0135 (5.8) b Paschen lines

Eq. p0135 (5.8)

Generalized energylevels in the hydrogen atom

Eq. p0270 Cross section for bound-free ionization of hydrogen in the n'th quantum state

Number of atoms pr cm^3


Number of illuminated lines in the grating


Number of particles/atoms


Number of steps


Order of diffraction lines


Side 25 af 36

Time difference in years (Epoch)



Total number density pr volume

Eq. p0225 (8.1)



Column density of neutral hydrogen (in unit 1/cm)


Number of atoms in ionization state i

Eq. p0234 (8.7) Saha

Index of refraction at wavelength lambda


Number of particles moving at the speed v

Eq. p0225 (8.1)


Frequency

Eq. p0081 a Frequency


Eq. p0131 c Frequency - wave length

Momentum


Side 26 af 36

Paralax in arcseconds

Eq. p0064 (3.1) Distance from paralax in arcseconds

Eq. p0064 a Distance in AU from parallax in arc seconds

Period

Eq. p0009a Synodic period of an inferior planet

Eq. p0009b Synodic period of superior planet


Eq. p0039 c Tangential speed

Eq. p0052 (2.35) b



Eq. p0211 (7.8)



Eq. p0210 (7.8) a



Pressure

Eq. p0323 (10.14)



Eq. p0328 (10.26)




Eq. p0352 (10.63)




Side 27 af 36

Eq. p0353 (10.65)



Eq. p0411 (11.8) Pressure structure of corona

Pressure of free electrons

Eq. p0234 (8.7) Saha

Radiation pressure

Eq. p0261 (9.9) Radiation pressure

Nuclear energy generated pr particle


Orbit radius




Eq. p0052 (2.35) b




Radius of/inside star/planet





Side 28 af 36

Eq. p0318 (10.7)








Eq. p0411 a Approximate ratio of proton potential and

kinetic energy at radius r0

Eq. p0411 (11.8) Pressure structure of corona

Distance to center of mass of object 1 (known)


Radius of curvature of surface 1 (known)


Distance to center of mass of object 2 (unknown)


Radius of curvature of surface 2 (unknown)


Distance at aphelion


Jeans radius


Side 29 af 36

Distance at perihelion


Density




Eq. p0280 (9.26) Pure emission



Eq. p0318 (10.7)




Eq. p0323 (10.14)



Eq. p0328 (10.26)




Eq. p0352 (10.63)








Eq. p0451 (12.16)


Distance traveled

Side 30 af 36


Synodic period

Eq. p0009a Synodic period of an inferior planet

Eq. p0009b Synodic period of superior planet

Cross section for bound-free photoionization

Eq. p0270

Cross section for bound-free ionization of hydrogen in the n'th quantum state

Collision cross section


Temperature / Effective temperature (Kelvin)

Eq. p0076 (3.15) Wiens displacement law



Eq. p0077 (3.18) Surface Flux of a star


Eq. p0225 (8.1)



Eq. p0229 (8.3)

Root-mean-squared speed of thermal particles

Eq. p0287 (9.37) Radiative flux

Eq. p0261 (9.9) Radiation pressure

Eq. p0323


Side 31 af 36

(10.14) Ideal gas - The ideal gaslaw applies

Eq. p0328 (10.26)





Eq. p0353 (10.65)




Eq. p0411 a

Approximate ratio of proton potential and kinetic energy at radius r0




Eq. p0234 (8.7) Saha


Eq. p0451 (12.16) Free-fall time for a homologous collapse

Optical depth of 21 cm line center


Optical depth


Eq. p0267 (9.16)


Incident angle

Side 32 af 36



Smallest angular seperation


Apparent ultraviolet magnitude


Inclination of rotational axis


Internal energy

Eq. p0353 (10.65) Internal energy Constraints:


Potential energy

Eq. p0056 (2.45) The virial theorem

Eq. p0056 (2.46) Total mechanical energy of system

Apparent visual magnitude


Orbital velocity


Speed


Eq. p0129 a Speed relative to the sun

Side 33 af 36

Eq. p0225 (8.1)


Speed of stellar wind


Speed of star 1 (known) in binary system


Speed of star 2 (unknown) in binary system


Velocity at aphelion


Rotational speed at equator


Escape velocity


Most probable speed


Velocity at perihelion


Radial velocity of object 1 (known)

Eq. p0211 The mass function of a binary system m1

Side 34 af 36

(7.8) comparable to m2



Eq. p0210 (7.8) a



Radial velocity of object 2 (unknown)



Radial velocity

Eq. p0110 (4.39) a Redshift


Root mean-squared speed of particles

Eq. p0229 (8.3) Root-mean-squared speed of thermal particles

Tangential velocity




Hydrogen Mass fraction


Eq. p0326 (10.21)


Side 35 af 36

Helium mass fraction


Eq. p0326 (10.21)


Metal mass fraction


Eq. p0326 (10.21)


Redshift

Eq. p0110 (4.39) a Redshift


Partition function

Eq. p0234 (8.7) Saha

Side 36 af 36

astronomy formulae

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