Time: 1 year = 3.156 x 10

Hubble time: H

Solar mass: 1 M

Proton mass: 1 m

Mass density: 1 M

Newton's constant:
G = 6.67 x 10^{-8} cm^{3} s^{-2} g^{-1} =
1.327 x 10^{11} km^{3} s^{-2} M_{⊙}^{-1} =
4.3 x 10^{-6} kpc km^{2} s^{-2} M_{⊙}^{-1}

Critical density of the universe:
ρ_{crit} = (3H_{0}^{2})/(8πG)

For H_{0} = 73 km s^{-1} Mpc^{-1},
ρ_{crit} = 10^{-29} g cm^{-3} =
1.5 x 10^{-7} M_{⊙} pc^{-3}

Speed: 1 km/s = 1.023 pc/Myr ≈ 1 kpc/Gyr

Frequency: 1 km/s/kpc ≈ 1 per Gyr

Acceleration: 1 (km/s)

**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.

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 m

**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,

cos^{2}(i) = [(b/a)^{2}-q^{2}]/[1-q^{2}],

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.

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: M_{HI} = 2.36 x 10^{5} D^{2} F_{HI}.

gives the mass of atomic hydrogen in M_{⊙} given
D, the distance in Mpc and F_{HI}, the
21 cm flux integral in Jy km s^{-1}.

**H _{2}:**
CO flux-to-H

gives the mass of molecular hydrogen in M

This is taken from Sheth et al (2005) who assume a CO-to-H

**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:

M_{gas} = X^{-1}(M_{HI} +M_{H2})

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_{*}/M_{o})^{α}

where α = 0.22 and M_{o} = 1.5 x 10^{24} 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(M_{H2}) = 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).

with A ≈ 50 M

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/(a

Critical surface density

Σ_{*} = a_{0}/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
g_{ex} ≪ g_{in} ≪ a_{0}

The case of domination by an **external field**:

σ ≈ [(G_{eff} M)/(3 r_{1/2})]^{1/2}

where r_{1/2} is the 3D half-mass radius and
G_{eff} ≈ (a_{0}/g_{ex})G
with g_{ex} being the amplitude of the external acceleration
(i.e., of a dwarf satellite orbiting a giant host). Strictly speaking,
this approximation only holds when
g_{in} ≪ g_{ex} ≪ a_{0} and
g_{ex} 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.

**Adibatic check**:

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/M_{host})^{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.