## Some Useful Numbers

Disclaimer: Astronomy involves astronomical scales, which often leads to some rather intense unit conversions. I started this page as a personal reference for unit conversions I found myself oft-repeating. Now, when I search for certain things, Google often cites this page, which means it leads others here. This is fine, if these numbers are useful to you, that's great. However, all numbers are provided as-is, without warranty that there are no typos, so you should check them for yourself before using them in something important like a scientific publication. (Yes, it has happened that there have been typos and people have published papers based on them.)

#### Basic Numbers and Conversions

Parsecs: 1 pc = 3.086 x 1013 km = 206265 AU ; 1 pc3 = 2.94 x 1055 cm3
Time: 1 year = 3.156 x 107 s
Hubble time: H0-1 = 13.4 Gyr for H0 = 73 km s-1 Mpc-1
Solar mass: 1 M = 1.989 x 1030 kg
Proton mass: 1 mp = 1.6726 x 10-27 kg = 0.938 GeV c-2
Mass density: 1 M pc-3 = 37.96 GeV cm-3 = 6.77 x 10-23 g cm-3

Newton's constant: G = 6.67 x 10-8 cm3 s-2 g-1 = 1.327 x 1011 km3 s-2 M-1 = 4.3 x 10-6 kpc km2 s-2 M-1

Critical density of the universe: ρcrit = (3H02)/(8πG)
For H0 = 73 km s-1 Mpc-1, ρcrit = 10-29 g cm-3 = 1.5 x 10-7 M pc-3

#### Galactic Units

Sometimes it is useful to express quantities in "Galactic" units, e.g.,
Speed: 1 km/s = 1.023 pc/Myr ≈ 1 kpc/Gyr
Frequency: 1 km/s/kpc ≈ 1 per Gyr
Acceleration: 1 (km/s)2/kpc = 3.24 x 10-14 m s-2

Galactic Time Scales
Galactic Day: 10 Myr (OB star lifetime)
Galactic Week: 70 Myr (vertical oscillation period in solar neighborhood)
Galactic Month: 150 Myr (epicyclic oscillation period in solar neighborhood)
Galactic Year: 210 Myr (orbital period of the sun)
Galactic Epoch: 10 Gyr (approximate main sequence lifetime of solar mass star)

The use of days, weeks, etc. is tongue in cheek astro-humor.
I have chosen values within measurement uncertainties so that 1 week = 7 days and 1 year = 3 weeks. Such an integer accounting is a choice; it cannot be exact. A classic result in Galactic dynamics is that the Galactic year is certainly not an integer multiple of the epicyclic period, so this definition of month is consistent with that feature of the terrestrial calendar even if there are fewer than two Galactic months in a Galactic year.

#### Photometry

Flux and apparent magnitude: conversion between linear flux f in physical units and the apparent magnitude:
m = −2.5log(f)+ξ
where ξ is the zero point specific to a particular filter in a given filter system. Historically, the Vega system is commonly used such that the star Vega has a magnitude mi = 0 in all filters i. This results in a different value of ξ for each filter. In contrast, the AB system sets ξ to the same constant for all filters.

Luminosity and absolute magnitude: conversion between linear L in solar luminosities L (the power emitted by the sun in a specified waveband) and absolute magnitude M:
M = −2.5log(L)+M
and vice-versa, in solar luminosities, L = 10[0.4(M − M)]
where M is the absolute magnitude of the sun in the relevant band (not to be confused with the solar mass M, for which the font should be distinguishable, but this page is more stable than html rendering browsers so sometimes it is not).

Surface brightness: conversion between linear Σ in L pc-2 and logarithmic μ in magnitudes arcsecond-2:
μ = 21.57 + M −2.5logΣ

Distance modulus: m-M = 5logD-5 when D is in pc; becomes 5logD+25 when D is in Mpc.
There are also terms for extinction and for the k-correction due to the redshifting of the source spectrum when the cosmic redshift becomes substantial, but you should either already know that or not be learning it from here.

Inclination: to infer the inclination of a disk from its image as projected on the sky,
cos2(i) = [(b/a)2-q2]/[1-q2],
where
i is the inclination (i = 0° is face-on; i = 90° is edge-on),
b/a is the observed minor/major axis ratio, and
q is the intrinsic thickness, i.e., the axis ratio when seen edge-on.
A common assumption is q = 0.15.

#### Gas Mass

The gaseous content of disk galaxies consists of atomic, molecular, and ionized components. Here I give the usual conversions from observed flux to mass for atomic and molecular gas, plus a scaling relation to estimate the total cold gas mass.
The mass of dust and ionized gas is typically negligible compared to the cold gas, at least within the optical disk. The circumgalactic medium surrounding galaxies may contain considerable amounts of ionized gas.

HI: HI flux-to-mass conversion: MHI = 2.36 x 105 D2 FHI.
gives the mass of atomic hydrogen in M given D, the distance in Mpc and FHI, the 21 cm flux integral in Jy km s-1.

H2: CO flux-to-H2-mass conversion: MH2 = 1.1 x 104 D2 FCO.
gives the mass of molecular hydrogen in M given D, the distance in Mpc and FCO, the J = 1 → 0 flux integral in Jy km s-1.
This is taken from Sheth et al (2005) who assume a CO-to-H2 conversion factor of XCO = 2.8 x 1020 cm-2 (K km/s)-1.

Cold Gas Mass: The formulae above refer to hydrogen mass only. To obtain the total cold gas mass, we must sum the atomic and molecular components, and correct them for the stuff that isn't hydrogen: helium and metals:

Mgas = X-1(MHI +MH2)

where X is the hydrogen mass fraction. To a first approximation, X = 3/4 (the primordial value). In more detail, metallicity increases with stellar mass; a useful scaling relation is

X = 0.75 - 38.2(M*/Mo)α

where α = 0.22 and Mo = 1.5 x 1024 M.

It often happens that direct tracers of the molecular gas mass (e.g., CO) are not available. In that case, the molecular gas mass can be approximated by

log(MH2) = log(M*) - 1.16.

The molecular gas mass is basically 7% of the stellar mass, with at least a factor of two intrinsic scatter. This isn't too bad, as the atomic gas usually outweighs the molecular gas by a comfortable factor.

If you use these scaling relations in a publication, please cite McGaugh et al. (2020).

#### Baryonic Tully-Fisher Relation

Mb = AVf4
with A ≈ 50 M km-4 s4
The above statement is purely empirical. See the data pages for updates on the calibration of A. In the particular case of MOND, A = X/(a0G) with the geometric factor X ≈ 0.8 and (a0G)-1 = 63 M km-4 s4. X here is a purely geometric quantity reflecting the fact that flattened disks rotate faster than the equivalent spherical mass distribution. It is not specific to MOND.

#### MOND acceleration constant

a0 = 1.2 x 10-10 m s-2 = 1.2 Å s-2 = 3700 km2 s-2 kpc-1 (about 2 furlongs fortnight-2).

Critical surface density
Σ* = a0/G = 860 M pc-2 = 1.8 kg m-2 (about the surface density of a stiff piece of construction paper).

MOND velocity dispersion estimators
for spherical systems with isotropic orbits in the deep MOND limit (the typical approximation applied to dwarf Spheroidals) depends on whether the object is isolated or subject to an external field that is comparable or stronger than its internal field. The formula that gives the smaller velocity dispersion is closer to being correct.

Isolated case:
σ = (M/1264)1/4
for σ in km/s and M in solar masses.
This applies when gex ≪ gin ≪ a0

The case of domination by an external field:
σ ≈ [(Geff M)/(3 r1/2)]1/2
where r1/2 is the 3D half-mass radius and Geff ≈ (a0/gex)G with gex being the amplitude of the external acceleration (i.e., of a dwarf satellite orbiting a giant host). Strictly speaking, this approximation only holds when gin ≪ gex ≪ a0 and gex is quasi-statiionary. The EFE is a dynamical effect; it waxes and wanes as a satellite orbits so these conditions are in practice rarely fixed in the way assumed by this simple formula. See also Haghi et al. who give useful fitting functions.

One can check if the static approximation is appropriate by using the adiabaticity parameter defined by Brada & Milgrom (2000):
γ = (D/r)3/2 (M/Mhost)1/2
which quantifies the number of orbits a star within the dwarf makes for every orbit the dwarf makes around its host. If γ is large, this implies that there is plenty of time for the system to adjust and application of the static EFE is likely adequate. If it is small, not so much. What is meant by small depends on the radius of the star within the dwarf; for the half light radius one would hope that γ > 2π (see McGaugh & Wolf 2010 for examples). Dwarfs that fail this check are subject to tidal disruption in MOND.
Note added: An example of failing to make this check is provided by Fattahi et al. Their claim "velocity dispersions are inconsistent with the predictions from Modified Newtonian Dynamics, a result that poses a possibly insurmountable challenge to that scenario" follows entirely from having failed to make this check. Ironically, they infer that these systems must be out of equilibrium in LCDM, which is the same interpretion found by McGaugh & Wolf for MOND. While there is no reason that they should have been aware of this web page, missing a previously published paper that addresses exactly this point was a major oversight that inverts the meaning of their interpretation.