constant value unit
speed of light in a vacuum c 2.99792458 x 108 m s-1
Planck's constant h 6.6260755 x 10-34 m2 kg s-1
Planck mass mp 2.17651 x 10-8 kg
Planck time mp 5.39121 x 10-44 s
Planck length p 1.61625 x 10-35 m
gravitational constant G 6.67259 x 10-11 m3 kg-1 s-2
electron mass me 9.1093897 x 10-31 kg
electron charge e 1.6021772 x 10-19 C
electron volt eV 1.6021772 x 10-19 J
proton mass mp 1.6726231 x 10-27 kg
neutron mass mn 1.6749286 x 10-27 kg
hydrogen mass mH 1.6733 x 10-27 kg
Avagadro's constant NA 6.02214 x 1023 n/a
Boltzmann constant K 1.3806488 x 10-23 m2 kg s-2 K-1
Stefan-Boltzman constant σ 5.670373 x 10-8 Watts m-2 K-4
fine structure constant α 7.29735308 x 10-3 n/a
Rydberg constant R 10973731.6 m-1
permittivity of free space ε0 8.8541 x 10 -12 m -3 kg -1 s4 Amp2
permeability of free space μ0 1.2566 x 10 -6 m kg s-2 Amp-2
Hubble constant H0 ~71 (or 100h) (km/s) Mpc-1
Hubble time 1/H0 3.45 x 1017 s
Hubble length c/H0 4228 Mpc
Astronomical Unit AU 1.496 x 1011 m
parsec pc 3.086 x 1016 m
light year ly 9.4605 x 1015 m
solar mass M 1.9889 x 1030 kg
solar radius R 6.955 x 108 m
solar bolometric luminosity L 3.839 x 1033 ergs s-1
critical density ρc 1.88 x 10-29 h2 g cm2
matter density parameter Ωm 0.25 n/a (ρmc)
dark energy density parameter ΩΛ 0.75 n/a
mass density fluctuation amplitude σ8 0.9 n/a
unit 1 = conversion unit 2
steradian 3282.8 degrees
ergs 1.602 x 10-12 eV
ergs s-1 cm-2   1000 W m-2
ergs s-1 cm-2-1   (2.998 x 1018) / λ2 ergs s-1 cm-2 Hz-1
Jansky 1 x 10-23 ergs s-1 cm-2 Hz-1
AB magnitude -2.5 Log(F) - 48.594 F (ergs s-1 cm-2 Hz-1)
AB magnitude -2.5 Log(F) - 5 Log(λ) - 2.402 F (ergs s-1 cm-2-1)
year 365.244 days
day 1440 minutes
hour 3600 seconds
equation description
Parallax distance
$$D_{\rm{p}}=\frac{1}{\theta}$$
The distance to a source (in parsecs) is 1 divided by the Parallax angle. P is half the angle (θ) a star moves with respect to the background stars in observations taken six months apart.
Luminosity
$$L=F 4 \pi D_{\rm{L}}^{2}$$
The intrinsic power emitted by a source of measured flux (F) at Luminosity distance (DL).
Rydberg formula
$$\Delta E = C_{1} Z^{2}\left(\frac{1}{n^2_2}-\frac{1}{n^2_1}\right)$$
The energy difference between two quantum states (n1 and n2) of a hydrogen-like atom or ion - C1=13.6 eV (or 1 Rydberg) and Z=atomic number
Redshift equation
$$\lambda_{\rm{o}} = \left(1+z\right) \lambda_{\rm{e}}$$
The difference between the observed (λo) and emitted (λe) wavelength of emission from a source at redshift, z
Recessional Velocity
$$v_{\rm{r}} = H_0 D$$
Also know and the Hubble Law - The velocity with which a galaxy at distance, D, appears to be moving away from us - H0=Hubble constant.
Redshift Velocity
$$(1+z)=\frac{(1+v/c)^{1/2}}{(1-v/c)^{1/2}}$$
The recession velocity at redshift, z, assuming the redshift is produced by a linear doppler shift.
Hubble time
$$t_{\rm{H}}=\dfrac{1}{H_0}$$
The standard cosmological unit of time defined by the Hubble Parameter.
Luminosity Distance
$$ \frac{d_{\rm{L}}}{(1+z)}$$ $$=\dfrac{c}{H_0}\int^{0}_{z}{\frac{\mathrm{d}z'}{\sqrt{\Omega_m(1+z')}}}$$
The distance (defined by the source dimming) to a source determined by the cosmology of the Universe.
Black Body Emission, λ
$$ I_{\rm{\lambda}}(\lambda)=\frac{2hc^2\lambda^{-5}}{e^{(hc/ \lambda kT)}-1} $$
The intensity of black body emission at temperature, T, as a function of wavelength - h=Planck's constant, k=Boltzman's Constant, c=speed of light.
Wein's Law, λ
$$ \lambda_{\rm{max}}T=2.898 \times 10^{-3}$$
The point where the black body intensity (as a function of λ) is a maximum.
Black Body Emission, ν
$$ I_{\rm{\nu}}(\nu)=\frac{2hc^{-2}\nu^{3}}{e^{(hc/ \nu kT)}-1} $$
Same as above but as a function of frequency.
Wein's Law, ν
$$ \nu_{\rm{max}}=5.897 \times 10^{10} T$$
Same as above but as a function of frequency.
Luminosity (Black Body)
$$ L=4 \pi R^2 \sigma T^4$$
The intrinsic brightness of a source of radius, R, and temperature, T (assuming it emits as a black body)
Flux
$$F=\frac{L}{4 \pi D_{\rm{L}}^{2}}$$
The emission measured per unit area from a source at Luminosity distance (DL). The Luminosity of the source, L, is spread over a sphere with surface area 4πDL2
Magnitude
$$ m=-2.5\mathrm{log}(F)-K$$
The logarithmic scale used to define the brightness of an astronomical source of flux, F. K is a constant which varies depending on the Magnitude system used.
Distance Modulus
$$ m=M+5\mathrm{log}\left(D_\rm{L}/10\right)$$
How the magnitude of a source varies with distance, DL (pc). m=observed magnitude, M=Absolute magnitude(The magnitude of the source if it were at a distance of 10pc)
Radial Velocity (exoplanets)
$$ \dfrac{\left(m_\rm{p}\,\mathrm{sin} (i)\right)^3}{m_{\star}^2}=\dfrac{v_{\rm{max}}^3P}{2\pi G}$$
The maximum observed line os sight velcoity (vmax) of a star of mass (m*) when obited by planet of mass (mp) at a period, P, and inclination angle, i.
Planet Temperature
$$ T_\rm{p}=T_{\star}\frac{\left(R_{\star}/2a_{\rm{p}}\right)^{1/2}}{\left(1-A\right)^{-1/4}}$$
Temperature of a planet (Tp), with albedo, A, at an obital distance, ap, from a star of temperature, T* and radius, R*.
Interstellar dispertion
$$\Delta t=k\, n\, d\, \left(\nu^{-2}_1-\nu^{-2}_2\right)$$
The time delay (Δt) between different radio frequency emission (ν1 and ν2) when it travels through distance, d, of an interstellar medium (ISM) with electron density, n. k is a constant with value k=4.15x10 -3 when t is in seconds, n is in electrons per m-3, d is in parsecs and ν is in MHz.
Faraday Rotation
$$ \theta=\alpha\, n\, d\, B\, \nu^{-2}+\theta_\rm{p}$$
The rotation of the polorization plane (θ) of polarized light with intrisic polorization (θp) and frequency ν as it passes through distance, d, of an ISM with electron density, n and magnetic field strenth in the line of sight, B. α=4.18x10 10 when θ is in degrees, n is in electrons per m-3, d is in parsecs, ν is in MHz and B is in Tesla.
Virial Theorem
$$ 2KE=-PE$$
In a stable system the kinetic energy (KE) of a body is twice its potential energy (PE).
Kinetic Energy
$$ KE=\frac{m\,v^2}{2}$$
The kinetic energy of a body of mass, m, moving at velocity, v.
Gravitational Potential Energy
$$ PE=\dfrac{-G\,m\,M}{r}$$
The gravitational potential energy of a body of mass, m, at a distance, r, from a larger body of mass, M.
Virial Rotation
$$ v=\sqrt{\dfrac{G\,M}{r}}$$
The velocity, v, of a body moving in virilized circular orbit of radius, r, around a mass, M.
Elliptical orbit (velocity)
$$ v=\sqrt{G\,M\,\left(\dfrac{2}{r}-\dfrac{1}{a}\right)}$$
The velcity, v, of a body moving in an elliptical orbit of semi-major axis, a, around a mass, M, when r is the radial distance between the two objetcs. Planets in the solar system have elliptical orbits following Kepler's laws.
Elliptical orbit (period)
$$ P=2\pi\sqrt{\dfrac{a^3}{G\,M}}$$
The period of orbit, P, of a body moving in an elliptical orbit of semi-major axis, a, around a mass, M.
Hydrostatic Equilibrium
$$ \dfrac{dP}{dr}=-\dfrac{G\,M(r)\,\rho(r)}{r^2}$$
The equilibrium obtained in a star of density, ρ, when the outward thermal pressure, P, is equal to the inward gravitational potential from mass, M, at radius, r.
select a waveband ⤵
Wavelength [Å] Line
1033.82 O VI
1215.24 Lyα
1240.81 N V
1305.53 O I
1335.31 C II
1397.61 Si IV
1399.8 Si IV + O IV
1549.48 C IV
1640.4 He II
1665.85 O III
1857.4 Al III
1908.734 C III
2326.0 C II
2439.5 Ne IV
2799.117 Mg II
Wavelength [Å] Line
emission
3346.79 Ne V
3426.85 Ne VI
3646 H→2
3727.092 O II
3729.875 O II
3889.0 He I
4072.3 S II
4102.89
4341.68
4364.436 O III
4862.68
4932.603 O III
4960.295 O III
5008.240 O III
6302.046 O I
6365.536 O I
6529.03 N I
6549.86 N II
6564.61
6585.27 N II
6718.29 S II
6732.67 S II
Wavelength [Å] Line
absorption
3934.777 K-band
3969.588 H-band
4305.61 G-band
5176.7 Mg
5895.6 Na
8500.36 CaII
8544.44 CaII
8664.52 CaII
Wavelength [µm] Line
1.0938 Paγ
1.252 [S IX]
1.2818 Paβ
1.430 [S X]
1.8751 Paα
1.932 [S XI]
1.962 [Si VI]
2.321 [Ca VIII]
2.483 [Si VII]
3.935 [Si IX]
Wavelength [µm] Line
10.5 [SIV]
12.8 [NeII]
14.3 [NeV]
15.6 [NeIII]
18.7 [SIII]
25.9 [OIV]
26.0 [FeII]
33.48 [SIII]
34.8 [SiII]
Wavelength [µm] Line
51.82 [OIII]
52.93-53.35 [OH]
57.32 [NIII]
58.70 H20
63.18 [OI]
65.13-65.28 OH
66.44-67.09 H2O
75.38 H2O
79.12-79.18 OH
84.42-84.60 OH
88.36 [OIII]
100.91-100.98 H2O
108.07 H2O
119.23-119.44 OH
121.89 [NII]
145.53 [OI]
157.74 [CII]
163.12-163.40 OH
162.81 CO(16-15)
Wavelength [mm] Freq [GHz] Line
0.372 807 12CO(7-6)
0.434 691 12CO(6-5)
0.453 661 13CO(6-5)
0.520 576 12CO(5-4)
0.544 551 13CO(5-4)
0.867 345.795 12CO(3-2)
1.30 230.538 12CO(2-1)
2.60 115.271 12CO(1-0)
2.64 113.5 CN(1-0)
3.31 90.6 HNC(1-0)
3.36 89.2 HCO+ (1-0)
3.38 88.6 HCN(1-0)
3.59 83.5 CH+(1-0)
6.11 49.0 CS (1-0)
6.91 43.4 SiO (1-0)
211 1.420 HI 21cm