Mathematical Foundations of Galactic Missing Mass: Disk, Sphere, Density, Potential, and Radial Scaling
TL;DR: The missing mass problem appears when the mass inferred from galactic rotation curves exceeds the mass directly observed in stars, gas, and dust. Mathematically, this requires connecting surface density on a disk, volume density in three dimensions, gravitational potential, radial acceleration, and enclosed mass.
1. Radial coordinates and geometry
We distinguish two geometries:
- Disk geometry: visible galactic matter is mainly distributed in a thin rotating disk.
- Spherical geometry: dark or missing mass is often modeled as a roughly spherical halo.
Disk area element:
\[ dA = R\,dR\,d\phi \]
Spherical volume element:
\[ dV = 4\pi r^2\,dr \]
The same symbol \(r\) is often used for galactocentric radius, but the meaning depends on the geometry. In a disk, \(R\) is a cylindrical radius. In a halo, \(r\) is usually a spherical radius.
2. Visible mass on a galactic disk
The visible disk is often approximated by an exponential surface density:
\[ \Sigma(R)=\Sigma_0 e^{-R/R_d} \]
The mass of an annulus between \(R\) and \(R+dR\) is:
\[ dM_{\rm disk}=2\pi R\Sigma(R)\,dR \]
\[ dM_{\rm disk}=2\pi R\Sigma_0e^{-R/R_d}\,dR \]
The cumulative visible disk mass is therefore:
\[ M_{\rm disk}(R)=2\pi\int_0^R R’\Sigma_0e^{-R’/R_d}\,dR’ \]
\[ M_{\rm disk}(R)=2\pi\Sigma_0R_d^2 \left[ 1-e^{-R/R_d} \left( 1+\frac{R}{R_d} \right) \right] \]
At large radius:
\[ M_{\rm disk}(R)\rightarrow 2\pi\Sigma_0R_d^2 \]
The visible disk mass approaches a finite value.
3. Spherical mass and volume density
For a spherical mass distribution, the volume density \(\rho(r)\) determines the enclosed mass:
\[ M(r)=4\pi\int_0^r \rho(r’)r’^2\,dr’ \]
The inverse relation is:
\[ \rho(r)=\frac{1}{4\pi r^2}\frac{dM}{dr} \]
This relation is central to the missing mass problem. If the inferred mass grows linearly with radius, then the corresponding spherical density decreases as \(1/r^2\).
4. Dynamical mass from circular motion
For circular motion, gravitational acceleration satisfies:
\[ \frac{v(r)^2}{r}=\frac{GM(r)}{r^2} \]
Therefore:
\[ M_{\rm dyn}(r)=\frac{v(r)^2r}{G} \]
For a flat rotation curve:
\[ v(r)\approx v_0 \]
\[ M_{\rm dyn}(r)\approx \frac{v_0^2}{G}r \]
This gives the standard scaling:
\[ M_{\rm dyn}(r)\propto r \]
\[ \rho_{\rm dyn}(r)\propto \frac{1}{r^2} \]
5. Missing mass definition
The missing mass is the difference between dynamical mass and visible mass:
\[ M_{\rm miss}(r)=M_{\rm dyn}(r)-M_{\rm vis}(r) \]
For an exponential visible disk:
\[ M_{\rm miss}(r)= \frac{v(r)^2r}{G} – 2\pi\Sigma_0R_d^2 \left[ 1-e^{-r/R_d} \left( 1+\frac{r}{R_d} \right) \right] \]
For \(v(r)\approx v_0\):
\[ M_{\rm miss}(r)\approx \frac{v_0^2}{G}r – 2\pi\Sigma_0R_d^2 \left[ 1-e^{-r/R_d} \left( 1+\frac{r}{R_d} \right) \right] \]
At large radius, the disk term saturates, while the dynamical term continues to grow approximately as \(r\).
6. Gravitational potential in 3D
The Newtonian gravitational potential generated by a point mass is:
\[ \Phi(r)=-\frac{GM}{r} \]
The corresponding gravitational field is the radial derivative of the potential:
\[ g(r)=-\frac{d\Phi}{dr} \]
\[ g(r)=-\frac{GM}{r^2} \]
This explains the relationship between \(1/r\) and \(1/r^2\): the potential of a localized mass falls as \(1/r\), while the force or acceleration falls as \(1/r^2\).
7. Poisson equation
Mass density and gravitational potential are connected through Poisson’s equation:
\[ \nabla^2\Phi=4\pi G\rho \]
In spherical symmetry, this becomes:
\[ \frac{1}{r^2}\frac{d}{dr} \left( r^2\frac{d\Phi}{dr} \right) = 4\pi G\rho(r) \]
This equation links three quantities:
\[ \rho(r) \longrightarrow M(r) \longrightarrow \Phi(r) \longrightarrow v(r) \]
8. Potential of an extended 3D density
For a general 3D density distribution, the potential is:
\[ \Phi(\mathbf{x}) = -G\int \frac{\rho(\mathbf{x}’)}{|\mathbf{x}-\mathbf{x}’|} \,d^3x’ \]
The kernel \(1/|\mathbf{x}-\mathbf{x}’|\) is the mathematical origin of the \(1/r\) potential in three dimensions.
9. Potential of a thin disk
For a thin disk with surface density \(\Sigma(R’)\), the gravitational potential in the disk plane can be written as:
\[ \Phi(R) = -G \int_0^\infty \int_0^{2\pi} \frac{\Sigma(R’)R’\,dR’\,d\phi} {\sqrt{R^2+R’^2-2RR’\cos\phi}} \]
The distance between a field point at radius \(R\) and a source point at radius \(R’\) is:
\[ d(R,R’,\phi) = \sqrt{R^2+R’^2-2RR’\cos\phi} \]
The radial acceleration in the disk is obtained by differentiating the potential:
\[ g_R(R)=-\frac{\partial \Phi}{\partial R} \]
The rotation speed follows from:
\[ v^2(R)=R\,|g_R(R)| \]
10. Projection of a 3D interaction onto the disk
If an interaction propagates in three dimensions but is evaluated in the disk plane, the radial projection introduces a geometric factor. For two points in the disk separated by distance \(d\), the radial projection factor is:
\[ \cos\theta = \frac{R-R’\cos\phi}{d(R,R’,\phi)} \]
Thus a generic 3D radial kernel \(K(d)\), projected onto the disk, appears as:
\[ K_{\rm disk}(R,R’,\phi) = K(d) \frac{R-R’\cos\phi}{d} \]
For example, a Newtonian force-like kernel has:
\[ K(d)\propto \frac{1}{d^2} \]
\[ K_{\rm disk}\propto \frac{R-R’\cos\phi}{d^3} \]
An exponential 3D kernel can be written as:
\[ K(d)\propto e^{-d/\lambda} \]
\[ K_{\rm disk}\propto e^{-d/\lambda} \frac{R-R’\cos\phi}{d} \]
11. Exponential kernels in three dimensions
A purely exponential radial factor has the form:
\[ e^{-r/\lambda} \]
In three-dimensional field theory, an exponentially screened potential often appears in Yukawa-like form:
\[ \Phi_Y(r)\propto -\frac{e^{-r/\lambda}}{r} \]
The associated radial force contains both \(1/r^2\) and exponential terms:
\[ g_Y(r) = -\frac{d}{dr} \left( \frac{e^{-r/\lambda}}{r} \right) \]
\[ g_Y(r)\propto e^{-r/\lambda} \left( \frac{1}{r^2} + \frac{1}{\lambda r} \right) \]
This shows why exponential radial behavior in 3D is not independent of \(1/r\) geometry. The exponential controls attenuation, while \(1/r\) and \(1/r^2\) arise from three-dimensional spreading.
12. Radial scaling laws
The missing mass problem is strongly tied to radial scaling. Several important radial laws appear repeatedly:
| Quantity | Typical scaling | Meaning |
|---|---|---|
| Potential of point mass | \(\Phi(r)\sim 1/r\) | 3D Green function of gravity |
| Force of point mass | \(g(r)\sim 1/r^2\) | Derivative of \(1/r\) |
| Flat rotation velocity | \(v(r)\sim constant\) | Observed in outer galactic disks |
| Dynamical mass | \(M(r)\sim r\) | Required by flat rotation |
| Halo density | \(\rho(r)\sim 1/r^2\) | Gives \(M(r)\sim r\) |
| Exponential disk | \(\Sigma(R)\sim e^{-R/R_d}\) | Visible disk fades rapidly |
| Screened 3D potential | \(\Phi(r)\sim e^{-r/\lambda}/r\) | Exponential attenuation plus 3D spreading |
13. From density to rotation curve
For a spherical halo with density:
\[ \rho(r)\propto \frac{1}{r^2} \]
the enclosed mass is:
\[ M(r)=4\pi\int_0^r \rho(r’)r’^2dr’ \]
\[ M(r)\propto r \]
Then:
\[ v^2(r)=\frac{GM(r)}{r} \]
\[ v^2(r)\approx constant \]
\[ v(r)\approx constant \]
This is the mathematical bridge between a \(1/r^2\) halo density and a flat galactic rotation curve.
14. From disk mass to missing mass
The visible disk mass grows rapidly at first and then saturates:
\[ M_{\rm disk}(R)\rightarrow M_d \]
The dynamical mass inferred from a flat rotation curve keeps growing:
\[ M_{\rm dyn}(r)\propto r \]
Therefore the missing mass behaves approximately as:
\[ M_{\rm miss}(r)=M_{\rm dyn}(r)-M_{\rm disk}(r) \]
\[ M_{\rm miss}(r)\approx \frac{v_0^2}{G}r-M_d \]
At sufficiently large radius:
\[ M_{\rm miss}(r)\sim r \]
\[ \rho_{\rm miss}(r)\sim \frac{1}{r^2} \]
15. Mathematical warning: disk and sphere are not interchangeable
The equation
\[ M(r)=\frac{v(r)^2r}{G} \]
is exact for spherical symmetry. For a flattened disk, one should calculate the potential by integrating over the disk and then derive the radial acceleration. The spherical expression is often used as an effective approximation, especially when discussing the mass needed to support a given rotation curve.
16. Key equations summary
Disk surface density:
\[ \Sigma(R)=\Sigma_0e^{-R/R_d} \]
Disk mass element:
\[ dM_{\rm disk}=2\pi R\Sigma(R)dR \]
Visible disk mass:
\[ M_{\rm disk}(R)=2\pi\Sigma_0R_d^2 \left[ 1-e^{-R/R_d} \left( 1+\frac{R}{R_d} \right) \right] \]
Spherical volume mass:
\[ M(r)=4\pi\int_0^r\rho(r’)r’^2dr’ \]
Density from enclosed mass:
\[ \rho(r)=\frac{1}{4\pi r^2}\frac{dM}{dr} \]
Dynamical mass:
\[ M_{\rm dyn}(r)=\frac{v(r)^2r}{G} \]
Missing mass:
\[ M_{\rm miss}(r)=M_{\rm dyn}(r)-M_{\rm vis}(r) \]
Newtonian potential:
\[ \Phi(r)=-\frac{GM}{r} \]
Poisson equation:
\[ \nabla^2\Phi=4\pi G\rho \]
3D potential integral:
\[ \Phi(\mathbf{x}) = -G\int \frac{\rho(\mathbf{x}’)}{|\mathbf{x}-\mathbf{x}’|} d^3x’ \]
Thin disk potential:
\[ \Phi(R) = -G \int_0^\infty \int_0^{2\pi} \frac{\Sigma(R’)R’dR’d\phi} {\sqrt{R^2+R’^2-2RR’\cos\phi}} \]
Projected radial kernel:
\[ \cos\theta= \frac{R-R’\cos\phi} {\sqrt{R^2+R’^2-2RR’\cos\phi}} \]
Exponential 3D screened potential:
\[ \Phi_Y(r)\propto -\frac{e^{-r/\lambda}}{r} \]
Conclusion
The missing mass problem is a mathematical mismatch between two radial behaviors. The visible disk follows an exponential surface density and reaches a finite cumulative mass. The dynamical mass inferred from approximately flat rotation curves grows roughly linearly with radius. If interpreted as a spherical halo, this corresponds to a density decreasing approximately as \(1/r^2\). The equations of disk integration, spherical shells, gravitational potential, and radial projection provide the mathematical language needed to analyze this mismatch.