Part III: Galactic Dynamics

1. Gravitational Potentials

The gravitational potential is the cornerstone of galactic dynamics. All dynamical quantities — orbital velocities, escape speeds, equilibrium distribution functions — derive from it. Because real galaxies have complex shapes, we construct models from analytic potential-density pairs that approximate observed components (bulge, disk, halo).

1.1 Poisson’s Equation

The gravitational potential $\Phi(\mathbf{x})$ is related to the mass density $\rho(\mathbf{x})$ through Poisson’s equation, the fundamental field equation of Newtonian gravity:

In Cartesian coordinates this is $\frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho$. For spherically symmetric systems we use the radial Laplacian:

$$\frac{1}{r^2}\frac{d}{dr}\left(r^2 \frac{d\Phi}{dr}\right) = 4\pi G \rho(r)$$

This can be integrated once using Gauss’s theorem. The gravitational acceleration at radius $r$ depends only on the enclosed mass:

$$g(r) = -\frac{d\Phi}{dr} = \frac{G M(r)}{r^2}, \quad M(r) = \int_0^r 4\pi r'^2 \rho(r')\,dr'$$

Given a density profile $\rho(r)$, one obtains $M(r)$ by integration, then the potential by a second integration: $\Phi(r) = -\int_r^\infty \frac{G M(r')}{r'^2}\,dr'$, choosing $\Phi \to 0$ as $r \to \infty$.

1.2 Circular and Escape Velocities

The circular velocity is the speed of a particle on a circular orbit in the equatorial plane. For a spherically symmetric potential, balancing centripetal and gravitational accelerations gives:

The escape velocity is the minimum speed needed to reach infinity from radius $r$:

$$v_{\mathrm{esc}}(r) = \sqrt{-2\Phi(r)} = \sqrt{2\int_r^\infty \frac{GM(r')}{r'^2}\,dr'}$$

For a point mass $v_{\mathrm{esc}} = \sqrt{2}\,v_c$. For extended mass distributions the ratio varies with radius.

1.3 Point Mass (Keplerian) Potential

The simplest potential, generated by a point mass $M$ at the origin:

$$\Phi(r) = -\frac{GM}{r}, \quad v_c(r) = \sqrt{\frac{GM}{r}} \propto r^{-1/2}$$

The Keplerian decline $v_c \propto r^{-1/2}$ is the signature of a centrally concentrated mass. Its absence in observed galactic rotation curves (which remain flat at large radii) constitutes one of the strongest arguments for dark matter.

1.4 Plummer Sphere

The Plummer model, originally devised for globular clusters, is the simplest softened potential:

Here $b$ is the Plummer scale length. The density has a finite central core and falls off as $r^{-5}$ at large radii. The total mass is $M$. While too steep at large radii for galaxy modeling, the Plummer sphere remains useful for numerical simulations due to its simplicity and finite central density.

1.5 Hernquist Profile

The Hernquist (1990) model provides a more realistic bulge/elliptical galaxy profile. It reproduces the$r^{1/4}$ de Vaucouleurs surface-brightness law when projected:

The scale radius $a$ sets the transition between the inner cusp ($\rho \propto r^{-1}$) and the outer envelope ($\rho \propto r^{-4}$). The half-mass radius is $r_{1/2} = (1 + \sqrt{2})\,a \approx 2.414\,a$. The circular velocity peaks at $r_{\max} = a$ with$v_c^{\max} = \frac{1}{2}\sqrt{GM/a}$.

1.6 Navarro–Frenk–White (NFW) Profile

Cosmological N-body simulations show that dark matter halos follow a nearly universal density profile discovered by Navarro, Frenk & White (1996, 1997):

The characteristic density $\rho_s$ and scale radius $r_s$ are related to the virial mass $M_{\mathrm{vir}}$ and concentration parameter$c = r_{\mathrm{vir}}/r_s$. The enclosed mass is:

$$M(r) = 4\pi\rho_s r_s^3 \left[\ln\!\left(1 + \frac{r}{r_s}\right) - \frac{r/r_s}{1 + r/r_s}\right]$$

The gravitational potential:

$$\Phi(r) = -\frac{4\pi G \rho_s r_s^3}{r}\ln\!\left(1 + \frac{r}{r_s}\right)$$

The circular velocity is:

$$v_c^2(r) = \frac{G M(r)}{r} = \frac{4\pi G \rho_s r_s^3}{r}\left[\ln\!\left(1 + \frac{r}{r_s}\right) - \frac{r/r_s}{1 + r/r_s}\right]$$

The NFW profile has a central cusp with $\rho \propto r^{-1}$ and an outer slope $\rho \propto r^{-3}$. Unlike the Hernquist profile, the total mass diverges logarithmically, so the profile must be truncated at the virial radius. The concentration parameter typically ranges from $c \sim 5$ for cluster-mass halos to $c \sim 15$ for Milky-Way-mass halos.

1.7 Miyamoto–Nagai Disk

For disk-like mass distributions, the Miyamoto & Nagai (1975) potential provides an axisymmetric model that interpolates between a Plummer sphere (when the disk scale length vanishes) and the Kuzmin disk (when the vertical scale height vanishes):

Here $R$ is the cylindrical radius, $a$ is the disk scale length, and$b$ is the vertical scale height. The limiting cases are: $a = 0$ gives the Plummer sphere with scale length $b$; $b = 0$ gives the infinitely thin Kuzmin disk. Typical Milky Way models use $a \sim 6.5$ kpc, $b \sim 0.26$ kpc.

1.8 Composite Galaxy Models

Real galaxies are modeled by superposing potentials from multiple components. Because Poisson’s equation is linear, the total potential is simply the sum of individual contributions:

$$\Phi_{\mathrm{total}} = \Phi_{\mathrm{bulge}} + \Phi_{\mathrm{disk}} + \Phi_{\mathrm{halo}}$$

The total circular velocity then satisfies $v_c^2 = v_{c,\mathrm{bulge}}^2 + v_{c,\mathrm{disk}}^2 + v_{c,\mathrm{halo}}^2$. A standard Milky Way model might combine a Hernquist bulge ($M \sim 10^{10}\,M_\odot$,$a \sim 0.5$ kpc), a Miyamoto–Nagai disk ($M \sim 5 \times 10^{10}\,M_\odot$,$a \sim 3.5$ kpc), and an NFW halo ($M_{\mathrm{vir}} \sim 10^{12}\,M_\odot$,$c \sim 10$).

2. The Collisionless Boltzmann Equation

A galaxy contains $\sim 10^{11}$ stars, each moving under the smooth gravitational field generated by all the others. Two-body encounters are so rare that the relaxation time exceeds the Hubble time by orders of magnitude. The stellar system is therefore collisionless, and its evolution is governed not by the Boltzmann equation of kinetic theory with a collision term, but by the collisionless Boltzmann equation (CBE).

2.1 The Distribution Function

The state of a collisionless stellar system is described by the distribution function (DF)$f(\mathbf{x}, \mathbf{v}, t)$, defined such that:

$$f(\mathbf{x}, \mathbf{v}, t)\,d^3\mathbf{x}\,d^3\mathbf{v}$$

gives the number of stars in the phase-space volume element $d^3\mathbf{x}\,d^3\mathbf{v}$centered on $(\mathbf{x}, \mathbf{v})$ at time $t$. By construction$f \geq 0$ everywhere. The spatial density is obtained by integrating over velocities:

$$\rho(\mathbf{x}, t) = m \int f(\mathbf{x}, \mathbf{v}, t)\,d^3\mathbf{v}$$

where $m$ is the stellar mass (or mass-weighted for a mass spectrum). Similarly, mean velocities and velocity dispersions are moments of $f$.

2.2 Derivation of the CBE

In a collisionless system, the flow in 6D phase space $(\mathbf{x}, \mathbf{v})$ is incompressible (Liouville’s theorem). The number of stars in any phase-space volume is conserved along the flow. This gives the continuity equation in phase space:

$$\frac{\partial f}{\partial t} + \sum_{i=1}^3 \frac{\partial(f \dot{x}_i)}{\partial x_i} + \sum_{i=1}^3 \frac{\partial(f \dot{v}_i)}{\partial v_i} = 0$$

Using Hamilton’s equations, $\dot{x}_i = v_i$ and$\dot{v}_i = -\partial\Phi/\partial x_i$. Since $\partial v_i / \partial x_i = 0$and $\partial \dot{v}_i / \partial v_i = 0$ (the gravitational force depends on position, not velocity), the equation simplifies to the CBE:

Written more compactly using the total derivative along the phase-space flow:

$$\frac{Df}{Dt} = 0$$

This states that $f$ is constant along any orbit in the potential $\Phi$. This result is the phase-space analogue of fluid incompressibility. Coupled with Poisson’s equation, we have a self-consistent system: $f$ determines $\rho$ which determines$\Phi$ which governs the orbits along which $f$ is constant.

2.3 Jeans Theorem

Jeans theorem states: Any steady-state solution of the collisionless Boltzmann equation depends on the phase-space coordinates only through the integrals of motion. Conversely, any non-negative function of the integrals of motion is a steady-state solution.

For a spherical, isotropic system, the only classical integral is the energy:

$$E = \frac{1}{2}v^2 + \Phi(r)$$

so $f = f(E)$. For an axisymmetric system, $f$ may also depend on the$z$-component of angular momentum $L_z = R v_\phi$:

$$f = f(E, L_z)$$

For anisotropic spherical systems, $f = f(E, L)$ where $L = |\mathbf{L}|$. The strong Jeans theorem further states that, for a potential in which most orbits are regular, $f$is a function of three independent isolating integrals.

2.4 The Jeans Equations

Integrating the CBE over all velocities yields moment equations analogous to the Euler equation of fluid dynamics. These are the Jeans equations. In spherical symmetry, the radial Jeans equation is:

Defining the velocity anisotropy parameter:

$$\beta(r) = 1 - \frac{\overline{v_\theta^2} + \overline{v_\phi^2}}{2\overline{v_r^2}} = 1 - \frac{\sigma_t^2}{2\sigma_r^2}$$

where $\sigma_r^2 = \overline{v_r^2}$ and $\sigma_t^2 = \overline{v_\theta^2} = \overline{v_\phi^2}$(for isotropy, $\beta = 0$; for purely radial orbits, $\beta = 1$), the Jeans equation becomes:

$$\frac{d(\rho \sigma_r^2)}{dr} + \frac{2\beta \rho \sigma_r^2}{r} = -\rho\frac{d\Phi}{dr} = -\rho\frac{GM(r)}{r^2}$$

This is the fundamental equation for mass modeling of spheroidal systems. Given observed surface-brightness profiles (which give $\rho$ after deprojection) and kinematic data (line-of-sight velocity dispersions), one can infer the enclosed mass $M(r)$, provided the anisotropy $\beta(r)$ is known or assumed. The mass-anisotropy degeneracy is a central difficulty in the dynamics of elliptical galaxies.

2.5 Cylindrical Jeans Equations

For disk galaxies, the cylindrical coordinate Jeans equations are more appropriate. The radial equation in the midplane ($z=0$) reads:

$$\frac{\partial(\rho\overline{v_R^2})}{\partial R} + \frac{\partial(\rho\overline{v_R v_z})}{\partial z} + \frac{\rho}{R}\left(\overline{v_R^2} - \overline{v_\phi^2}\right) = -\rho\frac{\partial\Phi}{\partial R}$$

The vertical equation gives the force perpendicular to the disk:

$$\frac{\partial(\rho\overline{v_z^2})}{\partial z} + \frac{\partial(\rho\overline{v_R v_z})}{\partial R} + \frac{\rho\overline{v_R v_z}}{R} = -\rho\frac{\partial\Phi}{\partial z}$$

The vertical Jeans equation, applied near the Sun, allows measurement of the local dark matter density (Oort’s analysis of the vertical force $K_z$).

3. Galaxy Rotation Curves and Dark Matter

The rotation curve $v_c(R)$ of a spiral galaxy is perhaps the single most important observable in galactic dynamics. Measurements of HI 21-cm emission, H-alpha optical emission, and CO molecular lines have revealed that rotation curves remain approximately flat well beyond the optical extent of galaxies, providing the most direct kinematic evidence for massive dark matter halos.

3.1 Keplerian vs. Flat Rotation Curves

If the mass of a galaxy were concentrated within its visible extent, the rotation curve beyond the luminous edge would follow the Keplerian falloff:

$$v_c(r) = \sqrt{\frac{GM_{\mathrm{tot}}}{r}} \propto r^{-1/2}$$

Instead, observations show $v_c(r) \approx \mathrm{const}$ out to the last measured point, often at $r > 5\,R_d$ where $R_d$ is the disk scale length. A flat rotation curve implies that the enclosed mass grows linearly with radius:

$$v_c = \mathrm{const} \implies M(r) = \frac{v_c^2 r}{G} \propto r$$

This requires $\rho(r) \propto r^{-2}$ (an isothermal density profile) extending far beyond the visible galaxy. This invisible mass is the dark matter halo.

3.2 Contribution of Bulge, Disk, and Halo

The observed rotation curve is the quadrature sum of contributions from distinct mass components:

$$v_c^2(R) = v_{c,\mathrm{bulge}}^2(R) + v_{c,\mathrm{disk}}^2(R) + v_{c,\mathrm{gas}}^2(R) + v_{c,\mathrm{halo}}^2(R)$$

Bulge: Modeled as a Hernquist or de Vaucouleurs profile. Dominates the inner 1–2 kpc. The contribution rises steeply, peaks near the bulge scale radius, then declines.

Disk: For an exponential surface density $\Sigma(R) = \Sigma_0 \exp(-R/R_d)$, the rotation curve of an infinitesimally thin disk is given by the Freeman (1970) formula:

$$v_{c,\mathrm{disk}}^2(R) = 4\pi G \Sigma_0 R_d\, y^2 \left[I_0(y)K_0(y) - I_1(y)K_1(y)\right]$$

where $y = R/(2R_d)$ and $I_n, K_n$ are modified Bessel functions. This peaks at $R \approx 2.2\,R_d$.

Gas: Contributes $\sim 10\text{--}20\%$ of the disk mass, with a surface density profile that often extends beyond the stellar disk.

Dark matter halo: The NFW profile or the pseudo-isothermal sphere$\rho(r) = \rho_0 / [1 + (r/r_c)^2]$ are commonly used. The halo dominates at large radii and ensures the flat rotation curve.

3.3 Mass Decomposition

Rotation curve decomposition fits the observed $v_c(R)$ with parameterized models for each component. The procedure involves:

1. Obtain surface photometry in near-infrared bands (to trace stellar mass).
2. Determine the stellar mass-to-light ratio $\Upsilon_*$ (from population synthesis or as a free parameter).
3. Convert surface brightness to surface mass density: $\Sigma_*(R) = \Upsilon_* I(R)$.
4. Compute $v_{c,*}(R)$ from $\Sigma_*(R)$.
5. Model HI and H$_2$ gas contributions.
6. Attribute the residual to dark matter: $v_{c,\mathrm{DM}}^2 = v_c^2 - v_{c,*}^2 - v_{c,\mathrm{gas}}^2$.
7. Fit a halo profile (NFW, isothermal, Burkert, etc.) to the residual.

The maximum-disk hypothesis assumes the largest $\Upsilon_*$ consistent with the data, minimizing the dark matter contribution in the inner galaxy. Under maximum disk, the stellar disk accounts for about 85% of the total rotation velocity at $R = 2.2\,R_d$.

3.4 Evidence for Dark Matter from Rotation Curves

Key observational results include:

• Rubin & Ford (1970, 1980): Flat rotation curves of spiral galaxies extending to $\sim 30$ kpc.
• Bosma (1981): HI rotation curves extending to $\sim 2\,R_{25}$, consistently flat.
• Tully–Fisher relation: $L \propto v_c^4$, linking luminosity and rotation velocity, reflecting the dark-to-baryonic mass relation.
• Low surface brightness (LSB) galaxies: Dark matter dominated even in the inner parts, providing strong constraints on halo profiles.
• Core-cusp problem: LSB and dwarf galaxy rotation curves favor constant-density cores ($\rho \propto r^0$) while CDM predicts cusps ($\rho \propto r^{-1}$).

The Burkert profile, $\rho(r) = \rho_0 / [(1 + r/r_0)(1 + (r/r_0)^2)]$, provides a better fit to many observed dwarfs than NFW, though baryonic feedback processes may modify the inner profiles of CDM halos to produce cores from cusps.

4. Disk Dynamics and Spiral Structure

Galactic disks are dynamically cold systems where stars move on nearly circular orbits with small random deviations. The interplay between differential rotation, self-gravity, and pressure-like velocity dispersion gives rise to rich dynamical phenomena: epicyclic oscillations, spiral arms, bars, and warps.

4.1 Epicyclic Motion

Consider a star on a nearly circular orbit in an axisymmetric disk potential. Linearizing the equations of motion about the circular orbit at radius $R_0$ (the guiding center), the radial perturbation $x = R - R_0$ obeys:

$$\ddot{x} = -\kappa^2 x$$

where the epicyclic frequency $\kappa$ is:

where $\Omega(R) = v_c/R$ is the circular frequency. The azimuthal perturbation is:

$$y = -\frac{2\Omega}{\kappa}x_0\sin(\kappa t + \phi_0)$$

The star traces an ellipse in the co-rotating frame with axis ratio $y_0/x_0 = 2\Omega/\kappa$. Key limiting cases:

• Solid-body rotation ($\Omega = \mathrm{const}$): $\kappa = 2\Omega$. The epicyclic ellipse has axis ratio 1:1 (a circle).
• Flat rotation curve ($v_c = \mathrm{const}$): $\kappa = \sqrt{2}\,\Omega$. The axis ratio is $\sqrt{2}:1$.
• Keplerian ($v_c \propto r^{-1/2}$): $\kappa = \Omega$. The epicycle closes in one orbital period (elliptical orbit).

4.2 Toomre Stability Criterion

A self-gravitating disk is subject to axisymmetric (ring-like) instabilities when self-gravity overwhelms the stabilizing effects of rotation (Coriolis force/epicyclic motion) and pressure (velocity dispersion). Toomre (1964) showed that the stability criterion for a thin stellar disk is:

For a gas disk, the analogous criterion uses the sound speed $c_s$ instead of $\sigma_r$, and the numerical coefficient changes:

$$Q_{\mathrm{gas}} = \frac{c_s \kappa}{\pi G \Sigma_{\mathrm{gas}}} > 1$$

The derivation proceeds by analyzing the dispersion relation for radial perturbations of wavelength$\lambda = 2\pi/k$ in a razor-thin disk. The dispersion relation is:

$$\omega^2 = \kappa^2 - 2\pi G \Sigma |k| + \sigma_r^2 k^2$$

Instability requires $\omega^2 < 0$. The most unstable wavenumber is$k_{\mathrm{crit}} = \pi G \Sigma / \sigma_r^2$, which gives the critical wavelength:

$$\lambda_{\mathrm{crit}} = \frac{2\sigma_r^2}{G\Sigma} = \frac{4\pi^2 G \Sigma}{\kappa^2}$$

Perturbations with wavelengths near $\lambda_{\mathrm{crit}}$ grow fastest when $Q < 1$. In practice, observed disk galaxies maintain $Q \sim 1.5\text{--}2$, marginally stable. The Toomre parameter regulates star formation: regions with $Q \lesssim 1$ fragment and form stars, which heats the disk and raises $Q$ back above unity (self-regulation).

4.3 Density Wave Theory (Lin–Shu)

The grand-design spiral structure seen in many disk galaxies (e.g., M51, M81) cannot be material arms: differential rotation would wind them up within a few orbits (the winding problem). The Lin & Shu (1964) density wave theory proposes that spiral arms are quasi-stationary wave patterns rotating at a fixed angular pattern speed $\Omega_p$ that differs from the material rotation speed$\Omega(R)$.

The dispersion relation for tightly wound (WKB) spiral waves in a stellar disk is:

$$(\omega - m\Omega)^2 = \kappa^2 - 2\pi G \Sigma |k| + \sigma_r^2 k^2$$

where $m$ is the number of arms and $\omega = m\Omega_p$ is the pattern frequency. Waves propagate where $(\omega - m\Omega)^2 > 0$, i.e., outside the Lindblad resonances.

Stars passing through the density crest are slowed (decelerated by the spiral gravitational field), spending more time in the arm, which enhances the density contrast. Gas is compressed as it enters the arm, triggering star formation, which produces the blue spiral arms seen in optical images. The wave pattern rotates rigidly while material flows through it.

4.4 Lindblad Resonances

Resonances occur where the forcing frequency in the rotating frame matches natural frequencies of stellar motion:

For $m = 2$ (two-armed spirals), the condition becomes$\Omega \pm \kappa/2 = \Omega_p$. Density waves can propagate only between the ILR and OLR. At the Lindblad resonances, waves are absorbed or reflected, transferring angular momentum to stars.

The angular momentum flux of spiral density waves plays a crucial role in the secular evolution of disk galaxies. Waves carry angular momentum outward, driving inward mass flow that feeds bulge growth and central activity.

4.5 Bar Instability

Dynamically cold disks are prone to the bar instability, a global $m = 2$ non-axisymmetric mode. N-body simulations show that a self-gravitating disk with insufficient random motion develops a strong bar within a few rotation periods. The Ostriker–Peebles (1973) criterion states that a disk becomes bar-unstable when:

$$t = \frac{T_{\mathrm{rot}}}{|W|} \gtrsim 0.14$$

where $T_{\mathrm{rot}}$ is the kinetic energy of ordered rotation and $W$is the gravitational potential energy. This criterion motivated the hypothesis that massive dark matter halos stabilize observed disks against bar formation (though many disk galaxies, including the Milky Way, do have bars).

Once formed, bars drive gas inward through gravitational torques, fueling nuclear starbursts and possibly AGN activity. The bar pattern speed $\Omega_b$ can be measured using the Tremaine–Weinberg method:

$$\Omega_b = \frac{\int h(x)\, \Sigma v_{\mathrm{los}}\, dx}{\int h(x)\, \Sigma x\, dx}$$

where the integral is along a slit parallel to the line of nodes and $h(x)$ is a weight function.

5. Milky Way Structure and Kinematics

Our Galaxy provides the most detailed laboratory for galactic dynamics. We can measure individual stellar positions, velocities (proper motions and radial velocities), and chemical abundances, enabling a level of dynamical analysis impossible for external galaxies. The Gaia mission has revolutionized this field by providing six-dimensional phase-space information for over a billion stars.

5.1 Structural Components

The Milky Way comprises several distinct dynamical components:

The dark matter halo extends to the virial radius $r_{\mathrm{vir}} \sim 200$ kpc, with a total virial mass $M_{\mathrm{vir}} \sim 1\text{--}1.5 \times 10^{12}\,M_\odot$ as determined from satellite kinematics, the timing argument, and escape velocity measurements.

5.2 Galactic Coordinates and Solar Motion

Galactic coordinates $(l, b)$ are defined with the Galactic center at$(l, b) = (0°, 0°)$ and the North Galactic Pole at $b = 90°$. The Sun is located at distance $R_0 = 8.178 \pm 0.026$ kpc from the Galactic center (GRAVITY Collaboration 2019).

The Local Standard of Rest (LSR) moves on a circular orbit with velocity$v_0 = \Theta_0 = 220\text{--}240$ km/s. The Sun’s peculiar velocity relative to the LSR is:

$$(U_\odot, V_\odot, W_\odot) \approx (11.1, 12.2, 7.3) \text{ km/s}$$

where $U$ is radially inward, $V$ is in the direction of Galactic rotation, and $W$ is toward the North Galactic Pole. The asymmetric drift$v_a = \Theta_0 - \overline{v_\phi}$ increases with velocity dispersion (hotter populations lag behind circular rotation).

5.3 Oort Constants

The Oort constants $A$ and $B$ describe the local velocity field of Galactic rotation as seen from the Sun. For a star at small distance $d$ and Galactic longitude $l$, the radial velocity and proper motion are:

$$v_r = Ad\sin 2l, \quad \mu_l = A\cos 2l + B$$

The Oort constants are defined as:

From these we derive local dynamical quantities:

• Local angular velocity: $\Omega_0 = A - B \approx 30$ km/s/kpc
• Local circular velocity: $v_0 = (A - B)R_0 \approx 240$ km/s
• Epicyclic frequency: $\kappa_0 = \sqrt{-4B(A-B)} \approx 37$ km/s/kpc
• Local shear rate: $A \approx 15.3$ km/s/kpc
• Local vorticity: $B \approx -11.9$ km/s/kpc (from Gaia DR3)

For a flat rotation curve, $dv_c/dR = 0$, so $A = -B = v_c/(2R)$ and$\kappa = \sqrt{2}\,\Omega$.

5.4 The Galactic Center: Sgr A* and Stellar Orbits

At the dynamical center of the Milky Way lies Sagittarius A* (Sgr A*), a compact radio source associated with a supermassive black hole. Decades of near-infrared monitoring of stellar orbits in the central arcsecond (the “S-stars”) have provided a precise mass measurement:

$$M_{\mathrm{BH}} = (4.154 \pm 0.014) \times 10^6\,M_\odot$$

The star S2 (S0-2) has an orbital period of only 16.05 years, a semi-major axis of$\sim 1000$ AU, and a pericenter distance of $\sim 120$ AU ($\sim 1400\,r_s$ where $r_s$ is the Schwarzschild radius). Its orbit has confirmed general relativistic effects including:

• Gravitational redshift at pericenter (GRAVITY Collaboration 2018)
• Schwarzschild precession of the orbit (GRAVITY Collaboration 2020)

The pericenter velocity of S2 reaches $\sim 7700$ km/s ($\sim 2.6\%$ of the speed of light), producing a combined gravitational and transverse Doppler shift:

$$\frac{\Delta \lambda}{\lambda} = \frac{GM_{\mathrm{BH}}}{c^2 r} + \frac{v^2}{2c^2}$$

The Galactic center also hosts a nuclear star cluster ($M \sim 2.5 \times 10^7\,M_\odot$), young massive stars in organized disk-like structures, and a circumnuclear disk of molecular gas. The dynamical interplay between the central black hole, the nuclear cluster, and infalling gas is a rich area of ongoing research.

5.5 Vertical Structure and the $K_z$ Problem

The vertical force perpendicular to the Galactic plane, $K_z = -\partial\Phi/\partial z$, constrains the local mass density. For a self-gravitating isothermal sheet:

$$\rho(z) = \rho_0\,\mathrm{sech}^2\!\left(\frac{z}{2h}\right), \quad h = \frac{\sigma_z}{\sqrt{2\pi G \rho_0}}$$

Applying the vertical Jeans equation and Poisson’s equation:

$$K_z(z) = -\frac{1}{\rho}\frac{\partial(\rho\sigma_z^2)}{\partial z} \quad \Rightarrow \quad \frac{\partial K_z}{\partial z} = -4\pi G \rho_{\mathrm{total}}$$

The total midplane density measured this way is$\rho_0 \approx 0.1\,M_\odot\,\mathrm{pc}^{-3}$, of which known baryonic matter accounts for about $0.09\,M_\odot\,\mathrm{pc}^{-3}$. The small residual constrains the local dark matter density to $\rho_{\mathrm{DM}} \sim 0.01\,M_\odot\,\mathrm{pc}^{-3} \approx 0.4$GeV/cm$^3$, a crucial input for direct dark matter detection experiments.

6. Dynamical Friction and Galaxy Interactions

When a massive body moves through a sea of lighter particles, the gravitational focusing of background particles behind it creates a density enhancement (a “wake”) that exerts a retarding force. This phenomenon, dynamical friction, is the engine of galaxy mergers, satellite orbital decay, and the growth of massive galaxies through hierarchical structure formation.

6.1 Chandrasekhar Dynamical Friction Formula

Chandrasekhar (1943) derived the deceleration of a body of mass $M$ moving at velocity$\mathbf{v}_M$ through an infinite, uniform, isotropic background of particles with density $\rho$ and velocity distribution $f(\mathbf{v})$:

where $X = v_M / (\sqrt{2}\,\sigma)$, $\sigma$ is the 1D velocity dispersion of the background, and $\ln\Lambda$ is the Coulomb logarithm:

$$\ln\Lambda = \ln\left(\frac{b_{\max}}{b_{\min}}\right) \approx \ln\left(\frac{r_{\mathrm{system}}}{GM/\sigma^2}\right)$$

Key properties of dynamical friction:

• The force is proportional to $M^2$ (one factor from gravitational focusing, one from the mass being decelerated).
• For $v_M \ll \sigma$: $F_{\mathrm{df}} \propto v_M$ (linear drag).
• For $v_M \gg \sigma$: $F_{\mathrm{df}} \propto v_M^{-2}$ (decreasing friction at high speeds).
• Only particles slower than $M$ contribute to the drag (faster particles are deflected but produce no net retardation).
• The deceleration is independent of the mass of background particles (it depends on $\rho$, not individual masses).

6.2 Orbital Decay Timescale

For a satellite of mass $M_{\mathrm{sat}}$ on a circular orbit at radius $r$in an isothermal halo with $v_c = \sqrt{2}\,\sigma = \mathrm{const}$, the dynamical friction timescale for orbital decay to the center is:

$$t_{\mathrm{df}} \approx \frac{1.17}{\ln\Lambda}\frac{M_{\mathrm{halo}}(r)}{M_{\mathrm{sat}}}\frac{r}{v_c} \approx \frac{1.17}{\ln\Lambda}\frac{v_c^2 r}{G M_{\mathrm{sat}}}r$$

Numerically, for a satellite of mass $10^{10}\,M_\odot$ at $r = 50$ kpc in a Milky Way-mass halo ($v_c \sim 220$ km/s, $\ln\Lambda \sim 3$),$t_{\mathrm{df}} \sim 5$ Gyr. This explains why the Magellanic Clouds, Sagittarius dwarf, and other satellites are still in the process of spiraling inward.

6.3 Tidal Stripping and the Tidal Radius

A satellite orbiting within a host galaxy experiences tidal forces that strip material beyond the tidal (Jacobi) radius. For a satellite of mass $m$ on a circular orbit at distance $D$from a host of mass $M$:

This is derived by equating the tidal acceleration from the host to the self-gravitational acceleration of the satellite at its surface:

$$\frac{2GMr_t}{D^3} = \frac{Gm}{r_t^2}$$

Material outside $r_t$ is stripped and forms tidal tails. Two tidal tails emerge: a leading tail (moving ahead on a lower-energy orbit) and a trailing tail (falling behind on a higher-energy orbit). These tidal streams are powerful probes of the host galaxy’s gravitational potential. Notable examples include:

• Sagittarius stream: Wraps around the Milky Way, constraining the shape of the dark matter halo.
• Palomar 5 tails: Extending $> 10°$ from a disrupting globular cluster.
• GD-1 stream: A thin, cold stream revealing perturbations from dark matter subhalos.

6.4 Galaxy Mergers

Galaxy mergers are classified by the mass ratio $q = M_2/M_1 \leq 1$:

• Major mergers ($q > 1/3$): Dramatically transform galaxy morphology. Two spirals can merge to form an elliptical (the “merger hypothesis” of Toomre & Toomre 1972).
• Minor mergers ($1/10 < q < 1/3$): Thicken disks, build up bulges, create shells and streams.
• Micro-mergers ($q < 1/10$): Satellite accretion, contributing to stellar halo buildup.

Mergers are further classified by gas content:

• Dry mergers: Gas-poor (elliptical + elliptical). Produce slowly rotating, massive ellipticals. No new star formation.
• Wet mergers: Gas-rich (spiral + spiral). Trigger intense starbursts (ultraluminous infrared galaxies, ULIRGs), funnel gas to the center (AGN fueling), and may rebuild a disk if enough gas survives.

The merger rate evolves with redshift approximately as $(1+z)^{2\text{--}3}$, reflecting the hierarchical growth of structure in $\Lambda$CDM cosmology.

6.5 Violent Relaxation

During a galaxy merger or gravitational collapse, the potential fluctuates rapidly on the dynamical timescale. Lynden-Bell (1967) showed that in a time-varying potential, the energy of individual stars changes:

$$\frac{dE}{dt} = \frac{\partial \Phi}{\partial t}$$

This process, called violent relaxation, redistributes stellar energies on a crossing time ($\sim 10^8$ yr), far faster than two-body relaxation. Key properties:

• Independent of stellar mass (unlike two-body relaxation), so it does not produce mass segregation.
• Drives the system toward a configuration resembling the isothermal distribution, but never quite reaches it (incomplete violent relaxation).
• Produces the universal surface-brightness profiles observed in elliptical galaxies.
• The timescale is $t_{\mathrm{vr}} \sim t_{\mathrm{cross}} \sim R/\sigma \sim 10^8$ yr.

The Lynden-Bell distribution function, derived by maximizing entropy subject to the constraints of the CBE, takes the form:

$$f(E) = \frac{\eta}{e^{\alpha + \beta E} + \eta}$$

which resembles a Fermi–Dirac distribution with “degeneracy parameter” $\eta$arising from the exclusion principle in phase space (finite-density initial conditions cannot be compressed to infinite density). In the non-degenerate limit ($\eta \to \infty$), this reduces to the isothermal distribution $f \propto e^{-\beta E}$.

6.6 Phase Mixing and Chaotic Mixing

After violent relaxation ceases (when the potential becomes nearly stationary), phase mixing completes the process of relaxation. Stars on nearby but distinct orbits have slightly different orbital frequencies, causing initially localized phase-space structures to wind up into ever-thinner filaments. Although $f$ is constant along orbits (by the CBE), the coarse-grained distribution function (averaged over finite phase-space volumes) evolves toward a smooth equilibrium.

The timescale for phase mixing is:

$$t_{\mathrm{mix}} \sim \frac{2\pi}{\Delta\Omega} \sim \frac{P}{\Delta P / P}$$

where $\Delta\Omega$ is the spread in orbital frequencies across the phase-space structure. In chaotic regions of phase space (e.g., near resonance overlaps in triaxial potentials), chaotic mixing accelerates relaxation exponentially, with the Lyapunov timescale replacing the orbital period.

6.7 The Virial Theorem

For a self-gravitating system in steady state, the scalar virial theorem relates the total kinetic energy $T$ and potential energy $W$:

This is derived from the tensor virial theorem by taking the trace. Combining with the total energy$E = T + W$:

$$T = -E, \quad W = 2E$$

For an observed system, the virial mass estimator is:

$$M_{\mathrm{vir}} = \frac{\alpha \sigma^2 R_h}{G}$$

where $\sigma$ is the line-of-sight velocity dispersion, $R_h$ is the half-light radius, and $\alpha$ is a dimensionless constant of order 10 that depends on the density and orbital anisotropy profiles. This is the foundation of mass measurements for elliptical galaxies, globular clusters, and galaxy clusters.

Summary