The Mass of the Milky Way Disk as a Function of Radius

TL;DR

The visible mass of the Milky Way disk can be modeled as the sum of several components: the thin stellar disk, the thick stellar disk, atomic hydrogen gas HI, and molecular hydrogen gas H₂.

The most useful equation is:

\(M_{\mathrm{disk,visible}}(where r is the distance from the Galactic Center in kiloparsecs, or kpc.

For the stellar part of the disk, using commonly adopted Milky Way parameters from McMillan’s Galactic mass model, the mass inside radius r is:

[latex]M_{\mathrm{disk,stars}}(with r in kpc and mass in solar masses, M⊙.

This equation describes the visible stellar mass of the Milky Way disk as a function of distance from the Galactic Center.

Final Equation for the Visible Disk Mass

The visible disk of the Milky Way can be written as:

[latex]M_{\mathrm{disk,visible}}(The stellar part is the cleanest:

[latex]M_{\mathrm{disk,stars}}(Using numerical parameters:

[latex]M_{\mathrm{disk,stars}}(where:

r = distance from the Galactic Center in kpc
M_disk,stars = stellar disk mass inside radius r
M⊙ = one solar mass

The parameters used here come from McMillan’s 2017 Milky Way mass model, which gives a Solar radius R₀ = 8.20 ± 0.09 kpc, circular speed v₀ = 232.8 ± 3.0 km/s, and total stellar mass (54.3 ± 5.7) × 10⁹ M⊙.

The Milky Way Disk Is Built from Rings

A simple way to understand the mass equation is to imagine cutting the Galactic disk into many thin circular rings.

Each ring has:

[latex]\mathrm{circumference}=2\pi r\) \(\mathrm{width}=dr\) \(\mathrm{area}=2\pi r\,dr\)

If the surface mass density of the disk is Σ(r), then the mass of one thin ring is:

\(dM=2\pi r\,\Sigma(r)\,dr\)

The mass inside radius r is obtained by adding all rings from the center to r:

\(M(This is the basic mathematical idea behind the disk mass equation.

The Exponential Disk Equation

The stellar disk of the Milky Way is usually approximated by an exponential surface density:

[latex]\Sigma(r)=\Sigma_0 e^{-r/R_d}\)

where:

Σ₀ = central surface mass density
R_d = disk scale length
r = distance from the Galactic Center

The scale length R_d tells us how quickly the disk becomes less dense as we move outward.

Substituting this density into the ring equation gives:

\(M(Solving the integral gives:

[latex]M(This is the main equation used for the stellar disk.

Component 1 — The Thin Stellar Disk

The thin disk is the bright, flat, star-forming part of the Milky Way. It contains young stars, many Sun-like stars, gas, dust, and the spiral arms.

For the thin stellar disk:

[latex]\Sigma_{0,\mathrm{thin}}=896\,M_\odot\,\mathrm{pc}^{-2}\) \(R_{d,\mathrm{thin}}=2.50\,\mathrm{kpc}\)

Since:

\(1\,\mathrm{kpc}^2=10^6\,\mathrm{pc}^2\)

we write:

\(\Sigma_{0,\mathrm{thin}}=896\times10^6\,M_\odot\,\mathrm{kpc}^{-2}\)

The mass inside radius r is:

\(M_{\mathrm{thin}}(Therefore:

[latex]M_{\mathrm{thin}}(The total thin disk mass is obtained by taking r → ∞:

[latex]M_{\mathrm{thin,total}}\simeq3.52\times10^{10}M_\odot\)

Component 2 — The Thick Stellar Disk

The thick disk is older, more vertically extended, and more diffuse than the thin disk. Its stars move farther above and below the Galactic plane.

For the thick stellar disk:

\(\Sigma_{0,\mathrm{thick}}=183\,M_\odot\,\mathrm{pc}^{-2}\) \(R_{d,\mathrm{thick}}=3.02\,\mathrm{kpc}\)

So:

\(\Sigma_{0,\mathrm{thick}}=183\times10^6\,M_\odot\,\mathrm{kpc}^{-2}\)

The mass inside radius r is:

\(M_{\mathrm{thick}}(Therefore:

[latex]M_{\mathrm{thick}}(The total thick disk mass is:

[latex]M_{\mathrm{thick,total}}\simeq1.05\times10^{10}M_\odot\)

Stellar Disk Mass: Thin Disk + Thick Disk

Adding both stellar components gives:

\(M_{\mathrm{disk,stars}}(or:

[latex]M_{\mathrm{disk,stars}}(At very large radius:

[latex]M_{\mathrm{disk,stars}}(\infty)=3.52\times10^{10}+1.05\times10^{10}\) \(M_{\mathrm{disk,stars}}(\infty)\simeq4.57\times10^{10}M_\odot\)

So, in this model, the visible stellar disk of the Milky Way contains about:

45.7 billion solar masses

Component 3 — Atomic Hydrogen Gas, HI

The Milky Way disk also contains visible gas. The first major gas component is atomic hydrogen, written HI.

Unlike the stellar disk, the gas is not well described by a simple exponential disk. It has a central depression, or “hole,” so a better form is:

\(\Sigma_{\mathrm{gas}}(r)=\Sigma_0\exp\left(-\frac{R_m}{r}-\frac{r}{R_d}\right)\)

For HI:

\(R_{d,\mathrm{HI}}=7.0\,\mathrm{kpc}\) \(R_{m,\mathrm{HI}}=4.0\,\mathrm{kpc}\) \(M_{\mathrm{HI,total}}\simeq1.1\times10^{10}M_\odot\)

The mass inside radius r is:

\(M_{\mathrm{HI}}(This equation says: take the total HI mass and multiply it by the fraction of the HI disk contained inside radius r.

Component 4 — Molecular Hydrogen Gas, H₂

The second major gas component is molecular hydrogen, written H₂. This gas is more closely associated with cold clouds and star formation.

For H₂:

[latex]R_{d,\mathrm{H_2}}=1.5\,\mathrm{kpc}\) \(R_{m,\mathrm{H_2}}=12.0\,\mathrm{kpc}\) \(M_{\mathrm{H_2,total}}\simeq1.2\times10^9M_\odot\)

The mass inside radius r is:

\(M_{\mathrm{H_2}}(

Full Visible Disk Mass Equation

Combining stars and gas:

[latex]M_{\mathrm{disk,visible}}(Fully written:

[latex]M_{\mathrm{disk,visible}}(where:

r and R are in kpc
M is in M⊙

This equation gives the visible disk mass of the Milky Way inside a radius r, measured from the Galactic Center.

Example: Mass Inside the Sun’s Orbit

The Sun is located at approximately:

[latex]R_0\simeq8.2\,\mathrm{kpc}\)

Using only the stellar disk equation:

\(M_{\mathrm{disk,stars}}(<8.2)\simeq3.52\times10^{10}\left[1-e^{-8.2/2.50}\left(1+\frac{8.2}{2.50}\right)\right]+1.05\times10^{10}\left[1-e^{-8.2/3.02}\left(1+\frac{8.2}{3.02}\right)\right][/latex]

Numerically, this gives approximately:

[latex]M_{\mathrm{disk,stars}}(<8.2\,\mathrm{kpc})\simeq3.7\times10^{10}M_\odot[/latex]

So, inside the Sun’s orbit, the stellar disk already contains most of its total mass.

Why Use Rings?

The ring method is useful because a galaxy disk is not a sphere.

For a spherical object, the mass shell at radius r has area:

[latex]4\pi r^2\)

But for a thin disk, the mass is spread over circular rings:

\(dM=2\pi r\Sigma(r)\,dr\)

This is why disk mass equations look different from spherical mass equations.

In a disk:

mass comes from rings

In a sphere:

mass comes from shells

The Milky Way contains both disk-like and spherical components, but this page focuses on the disk.

What This Equation Includes

The equation includes:

Component	Meaning	Included?
Thin stellar disk	Young and intermediate-age stars near the Galactic plane	Yes
Thick stellar disk	Older stars farther from the plane	Yes
HI gas	Atomic hydrogen	Yes
H₂ gas	Molecular hydrogen	Yes
Bulge/bar	Central stellar structure	No
Dark matter halo	Invisible gravitational component	No
Stellar halo	Very diffuse old stars	No

This is why we call it the visible disk mass, not the full mass of the Milky Way.

How This Connects to the Missing Mass

Once the visible disk mass is known, astronomers compare it with the mass required by the observed rotation of the Galaxy.

The dynamical mass inferred from circular motion is:

\(\)M_{\mathrm{dyn}}(In practical units:

\(\)M_{\mathrm{dyn}}(The missing mass is then:

\(\)M_{\mathrm{missing}}(For this page, the disk contribution is:

\(\)M_{\mathrm{visible}}(A full Milky Way model would also add the central bulge/bar and other minor baryonic components.

Important Limitations

This model is useful, but it is not perfect.

First, the Milky Way is not a perfectly smooth axisymmetric disk. It has spiral arms, a central bar, star-forming regions, and local structures.

Second, gas is difficult to model because we observe it from inside the Galaxy. Its distance and rotation must be reconstructed from velocity data.

Third, the disk has vertical thickness. The equations above are mostly surface-density equations, which are excellent for radial mass profiles but do not describe every vertical detail.

Fourth, the parameters depend on the adopted Galactic model. McMillan’s model is a strong reference point, but different studies may give slightly different disk masses, scale lengths, and gas profiles. McMillan explicitly reports statistical uncertainties for key global parameters such as R₀, v₀, stellar mass, virial mass, and local dark-matter density.

Glossary

Galactic Center
The central region of the Milky Way, around the supermassive black hole Sagittarius A*.

Kiloparsec, kpc
A distance unit used in galactic astronomy. One kiloparsec is about 3,260 light-years.

Solar mass, M⊙
The mass of the Sun. It is used as the standard mass unit in astronomy.

Surface density, Σ(r)
Mass per unit area of the Galactic disk at radius r.

Scale length, R_d
The distance over which the disk density decreases by a factor of e.

Thin disk
The flat, dense, star-forming disk of the Milky Way.

Thick disk
An older, more vertically extended stellar disk surrounding the thin disk.

HI
Atomic hydrogen gas.

H₂
Molecular hydrogen gas.

Dynamical mass
The mass required to explain the observed orbital speed of stars and gas.

Missing mass
The difference between the dynamical mass and the visible mass.

Accessibility Notes

Suggested image alt text:

Alt text for diagram 1: “Face-on view of the Milky Way disk divided into circular rings around the Galactic Center.”
Alt text for diagram 2: “Side view of the Milky Way showing a thin stellar disk embedded inside a thicker, older stellar disk.”
Alt text for graph: “Graph showing cumulative visible disk mass increasing with radius from the Galactic Center.”

Use readable labels such as:

“Radius from Galactic Center, kpc”
“Mass inside radius, solar masses”
“Thin disk”
“Thick disk”
“Gas disk”
“Total visible disk”

Suggested External References

Visible mass

To estimate the visible mass of the Milky Way at any radius, choose a value of r in kpc and insert it into:

For a first calculation, use the simpler stellar-disk equation. Then add HI and H₂ gas for a more complete visible disk model.

The Mass of the Milky Way Disk as a Function of Radius

Final Equation for the Visible Disk Mass

The Milky Way Disk Is Built from Rings

The Exponential Disk Equation

Component 1 — The Thin Stellar Disk

Component 2 — The Thick Stellar Disk

Stellar Disk Mass: Thin Disk + Thick Disk

Component 3 — Atomic Hydrogen Gas, HI

Component 4 — Molecular Hydrogen Gas, H₂

Full Visible Disk Mass Equation

Example: Mass Inside the Sun’s Orbit

Why Use Rings?

What This Equation Includes

How This Connects to the Missing Mass

Important Limitations

Glossary

Accessibility Notes

Suggested Internal Links

Suggested External References

Visible mass