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 = 1.393 x 10-13 m s-2 pc2 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 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.
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:
Distance modulus: m-M =
5logD-5 when D is in pc; becomes 5logD+25 when D is in Mpc.
Inclination: to infer the inclination of a disk from its image as
projected on the sky,
HI:
HI flux-to-mass conversion: MHI = 2.36 x 105 D2 FHI.
H2:
CO flux-to-H2-mass conversion:
MH2 = 1.1 x 104 D2 FCO.
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).
Critical surface density
MOND velocity dispersion estimators
Isolated case:
The case of domination by an external field:
Adibatic check:
μ = 21.57 +
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.
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.
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.
HI surface brightness: 1 M⊙ pc-2 =
1.25 × 1020 H atoms cm-2.
This is just the mass of the sun in protons spread over a square parsec in cm.
See section 7.8 of
Essential Radio Astronomy.
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.
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).
Σ* = a0/G = 860 M⊙ pc-2
= 1.8 kg m-2 (about the surface density of a stiff piece of
construction paper).
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.
σ = (M/1264)1/4
for σ in km/s and M in solar masses.
This applies when
gex ≪ gin ≪ a0
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.