# Governing Equations Reference — Physically Rigorous 3D Solar System + Black Hole Simulator

**Global conventions.** SI units throughout unless noted: `G = 6.67430e-11 m^3 kg^-1 s^-2` (CODATA 2018), `c = 2.99792458e8 m/s` (exact), `hbar = 1.054571817e-34 J s`, `k_B = 1.380649e-23 J/K`, `sigma_SB = 5.670374419e-8 W m^-2 K^-4`, `M_sun = 1.989e30 kg`, `GM_sun = 1.32712440018e20 m^3/s^2` (IAU/DE ephemeris value — in practice always use the product `GM`, known far more precisely than `G` or `M` separately). `mu = G(M+m)` is the two-body gravitational parameter (`mu ≈ GM` for a test body). Angles in radians. Geometrized units (`G=c=1`) noted where standard; convert masses to lengths via `M_geo = GM/c^2`.

---

## A. Newtonian N-Body

### A.1 N-Body Gravitational Acceleration

$$\vec{a}_i = G \sum_{\substack{j=1 \\ j \neq i}}^{N} \frac{m_j\,(\vec{r}_j - \vec{r}_i)}{|\vec{r}_j - \vec{r}_i|^3}$$

```
a_i = G * sum_{j=1..N, j != i} [ m_j * (r_j - r_i) / |r_j - r_i|^3 ]
```

- `a_i` — acceleration vector of body i [m/s^2]
- `G` — gravitational constant [m^3 kg^-1 s^-2]
- `m_j` — mass of body j [kg]; `r_i, r_j` — position vectors in an inertial frame [m]; `N` — number of bodies
- Ephemeris convention: propagate `GM_j` products directly (e.g. `GM_sun = 1.32712440041279419e20 m^3/s^2` per DE440). Classical astrodynamics units: AU / days / solar masses with the Gaussian constant `k = 0.01720209895` (`G -> k^2`).

Sources: [Harvard Math 118 N-body notes](https://legacy-www.math.harvard.edu/archive/118r_spring_05/handouts/nbody.pdf) · [Klioner, Basic Celestial Mechanics](https://sirrah.troja.mff.cuni.cz/~davok/Sergei_Klioner_skripta.pdf) · [JPL SSD Astrodynamic Parameters](https://ssd.jpl.nasa.gov/astro_par.html)

### A.2 Total Mechanical Energy

$$E = \sum_{i=1}^{N} \tfrac{1}{2} m_i |\vec{v}_i|^2 \;-\; \sum_{i=1}^{N}\sum_{j=i+1}^{N} \frac{G\, m_i m_j}{r_{ij}}$$

```
E = sum_i( 0.5*m_i*v_i^2 ) - sum_{i<j}( G*m_i*m_j / r_ij ),   r_ij = |r_i - r_j|
```

- `E` — total energy [J], conserved for an isolated system; `v_i` — speed of body i [m/s]; `r_ij` — pair separation [m]
- Each unique pair counted once (`i<j`); equivalent form `-(1/2)*G*sum_i sum_{j!=i} m_i*m_j/r_ij` for independent loops.

Sources: [Harvard Math 118 N-body notes](https://legacy-www.math.harvard.edu/archive/118r_spring_05/handouts/nbody.pdf) · [Klioner, Basic Celestial Mechanics](https://sirrah.troja.mff.cuni.cz/~davok/Sergei_Klioner_skripta.pdf)

### A.3 Total Angular Momentum Vector

$$\vec{L} = \sum_{i=1}^{N} m_i \left(\vec{r}_i \times \vec{v}_i\right)$$

```
L = sum_i( m_i * cross(r_i, v_i) )
```

- `L` — total angular momentum [kg m^2/s], conserved for an isolated system (pairwise central forces, zero net internal torque); use barycentric origin for the frame-independent value.

Sources: [Harvard Math 118 N-body notes](https://legacy-www.math.harvard.edu/archive/118r_spring_05/handouts/nbody.pdf) · [LibreTexts (Cline), Angular Momentum of a Many-Body System](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Variational_Principles_in_Classical_Mechanics_(Cline)/02:_Review_of_Newtonian_Mechanics/2.09:_Angular_Momentum_of_a_Many-Body_System) · [Klioner notes](https://sirrah.troja.mff.cuni.cz/~davok/Sergei_Klioner_skripta.pdf)

### A.4 Barycenter and Total Momentum

$$\vec{R}_{\rm cm} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i}, \qquad \vec{P} = \sum_i m_i \vec{v}_i = M_{\rm tot}\,\vec{v}_{\rm cm}$$

```
R_cm = sum_i(m_i * r_i) / sum_i(m_i)
P    = sum_i(m_i * v_i)            # = M_tot * v_cm
```

- `R_cm` — barycenter position [m]; `P` — total momentum [kg m/s]; `M_tot` — total mass [kg]
- `P` conserved for an isolated system: the barycenter moves uniformly, defining the inertial barycentric frame (basis of the IAU/IERS BCRS used by ephemerides).

Sources: [Wikipedia — Center of mass](https://en.wikipedia.org/wiki/Center_of_mass) · [orbital-mechanics.space — Motion of the Barycenter](https://orbital-mechanics.space/the-n-body-problem/motion-of-the-barycenter.html) · [NASA Basics of Spaceflight glossary](https://science.nasa.gov/learn/basics-of-space-flight/glossary/)

### A.5 Escape Velocity

$$v_{\rm esc} = \sqrt{\frac{2GM}{r}}$$

```
v_esc = sqrt(2*G*M / r)
```

- `v_esc` — minimum unpowered speed to reach infinity with zero residual speed, from distance `r` [m] of mass `M` [kg]. From `(1/2)v^2 - GM/r = 0`.

Sources: [Wikipedia — Escape velocity](https://en.wikipedia.org/wiki/Escape_velocity) · [physicsfundamentals.org — Escape velocity](https://physicsfundamentals.org/blog/escape-velocity)

### A.6 Hill Sphere Radius

$$r_H \approx a\sqrt[3]{\frac{m}{3M}} \qquad\text{(pericenter refinement: } r_H \approx a(1-e)\sqrt[3]{m/3M}\text{)}$$

```
r_H = a * (m / (3.0*M))^(1/3)
```

- `r_H` — radius within which the secondary's gravity dominates over the primary's tide [m]; `a` — orbit semi-major axis [m]; `m` — secondary mass [kg]; `M` — primary mass [kg]; valid for `m << M`. Derived from the L1/L2 Lagrange points of the circular restricted three-body problem.

Sources: [Wikipedia — Hill sphere](https://en.wikipedia.org/wiki/Hill_sphere) (cites de Pater & Lissauer, *Planetary Sciences*; Hamilton & Burns 1992) · [astronomicalreturns.com — Hill spheres](https://www.astronomicalreturns.com/2022/08/hill-spheres-where-moonmoons-are.html) · [Physics Forums — Hill sphere derivation](https://www.physicsforums.com/threads/complete-hill-sphere-derivation-learn-the-equation-and-its-derivation-process.197797/)

### A.7 Laplace Sphere of Influence

$$r_{\rm SOI} = a\left(\frac{m}{M}\right)^{2/5}$$

```
r_SOI = a * (m/M)^(2/5)
```

- `r_SOI` — patched-conic sphere-of-influence radius [m]; `a` — secondary–primary distance [m]; `m, M` — secondary/primary masses [kg]
- **Hill vs. Laplace — not interchangeable:** Hill has exponent **1/3** with the factor 3 (`a(m/3M)^(1/3)`, a dynamical stability boundary from L1/L2); Laplace SOI has exponent **2/5**, no factor 3 (a perturbation-ratio patching criterion). For Earth: `r_SOI ≈ 9.24e5 km` vs `r_H ≈ 1.5e6 km`.

Sources: [orbital-mechanics.space — Sphere of Influence](https://orbital-mechanics.space/interplanetary-maneuvers/sphere-of-influence.html) · [poliastro docs — SOI](https://poliastro-py.readthedocs.io/en/latest/api/safe/threebody/soi.html) · [Wikipedia — Sphere of influence (astrodynamics)](https://en.wikipedia.org/wiki/Sphere_of_influence_(astrodynamics))

### A.8 Roche Limit (Rigid and Fluid)

$$d_{\rm rigid} = R_M\left(2\,\frac{\rho_M}{\rho_m}\right)^{1/3} \approx 1.26\, R_M\left(\frac{\rho_M}{\rho_m}\right)^{1/3}, \qquad d_{\rm fluid} \approx 2.44\, R_M\left(\frac{\rho_M}{\rho_m}\right)^{1/3}$$

```
d_rigid = R_M * (2.0*rho_M/rho_m)^(1/3)     # ~ 1.26 * R_M * (rho_M/rho_m)^(1/3)
d_fluid = 2.44 * R_M * (rho_M/rho_m)^(1/3)
```

- `d` — center-to-center disruption distance [m]; `R_M` — primary radius [m]; `rho_M, rho_m` — mean densities of primary and satellite [kg/m^3]
- Coefficient notes (cross-checked): `1.26 = 2^(1/3)` — static rigid body (self-gravity vs. tide only); adding the synchronous-rotation centrifugal term gives ≈1.44–1.45 (UMD ASTR630 derives 1.45); `2.44` — Roche's classical fluid equilibrium-ellipsoid value (precise treatments give 2.42–2.46). Rings persist inside, moons outside.

Sources: [UMD ASTR630 Roche limit handout](https://pages.astro.umd.edu/~dphamil/ASTR630/handouts/RocheLimit.pdf) · [mathscinotes.com — Roche limit examples](https://www.mathscinotes.com/2016/03/roche-limit-examples/) · [astronomicalreturns.com — Roche limit](https://www.astronomicalreturns.com/2021/06/roche-limit-radius-of-disintegration.html) · [Grokipedia — Roche limit](https://grokipedia.com/page/Roche_limit)

### A.9 Tidal Acceleration Across a Body

$$a_{\rm tidal} \approx \frac{2GM\,d}{r^3}$$

```
a_tidal = 2.0*G*M*d / r^3
```

- `a_tidal` — differential acceleration between two points separated by `d` [m] along the line to mass `M` [kg], at distance `r` [m] (`d << r`). Use `d` = body diameter for near/far-side stretch, or `d` = center-to-surface distance for stress relative to the freely falling center. First-order Taylor expansion of `GM/x^2`; transverse (compressive) component is ~half magnitude, opposite sign. Basis of Roche limit (A.8) and spaghettification estimates.

Sources: [Wikipedia — Tidal force](https://en.wikipedia.org/wiki/Tidal_force) · [UCSD Physics 161 lecture (Griest)](https://courses.physics.ucsd.edu/2011/Winter/physics161/p161.14feb11.pdf) · [CMU 33-331 tidal forces](https://www.andrew.cmu.edu/course/33-331/pm11.pdf) · [SFU PHYS 390 Lecture 8](https://www.sfu.ca/~boal/390lecs/390lec8.pdf)

---

## B. Keplerian Mechanics

### B.1 Kepler's Three Laws

- **Law I (Ellipses):** each bound orbit is an ellipse with the central body at one focus (general two-body result: any conic section — ellipse/parabola/hyperbola — per the sign of the orbital energy).
- **Law II (Equal Areas):** the focus–body line sweeps equal areas in equal times; areal velocity is constant and equals half the specific angular momentum.
- **Law III (Harmonies):** `T^2 ∝ a^3` with constant `4 pi^2 / (G(M+m))`.

$$\frac{dA}{dt} = \frac{1}{2}r^2\dot\theta = \frac{h}{2} = \text{const}, \qquad T^2 = \frac{4\pi^2}{G(M+m)}\,a^3$$

```
dA/dt = 0.5 * r^2 * dtheta_dt = h/2 = constant     # h = |cross(r, v)|
T^2   = (4*pi^2 / (G*(M+m))) * a^3
```

- `A` — swept area [m^2]; `r` — radius [m]; `h` — specific angular momentum [m^2/s]; `T` — period [s]; `a` — semi-major axis [m]; `M, m` — masses [kg]

Sources: [NASA Science — Orbits and Kepler's Laws](https://science.nasa.gov/solar-system/orbits-and-keplers-laws/) · [JPL fltops, Two Body Problem ch. 9 (eqs. 9.3.2, 9.2.2)](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/CelestialMechanics9-2%20body%20prob.pdf) · [Princeton Dynamics §3.4](https://www.princeton.edu/~wbialek/intsci_web/dynamics3.4.pdf)

### B.2 Classical Orbital Elements and Anomalies

```
elements = [a, e, i, Omega, omega, M0]    # M0 (or time of periapsis T_p) fixes phase at epoch
```

- **a — semi-major axis** [m]: half the major axis.
- **e — eccentricity** [–]: `e = c/a` (center–focus distance over a); 0 circular, 0<e<1 ellipse, 1 parabola, >1 hyperbola.
- **i — inclination** [rad, 0..pi]: angle between orbit plane and reference plane = angle between `h_vec` and reference pole; `i<90°` prograde, `i>90°` retrograde.
- **Omega — longitude of ascending node** [rad, 0..2pi]: in the reference plane, from the reference x-axis (vernal equinox) to the ascending node (south-to-north crossing). Undefined for equatorial orbits.
- **omega — argument of periapsis** [rad, 0..2pi]: in the orbit plane, from ascending node to periapsis, along the motion. Undefined for circular orbits.
- **nu — true anomaly** [rad]: physical focus-centered angle from periapsis to the body.
- **E — eccentric anomaly** [rad]: auxiliary angle at the ellipse **center** via the circumscribed circle of radius a; perifocal position `(a cos E - ae, a sqrt(1-e^2) sin E)`.
- **M — mean anomaly** [rad]: fictitious uniformly increasing angle, `M = n(t - T_p)`; the time variable, converted to `E`, `nu` via Kepler's equation.

Sources: JPL SSD glossary — [semi-major axis](https://ssd.jpl.nasa.gov/glossary/semimajor_axis.html), [eccentricity](https://ssd.jpl.nasa.gov/glossary/eccentricity.html), [inclination](https://ssd.jpl.nasa.gov/glossary/inclination.html), [node](https://ssd.jpl.nasa.gov/glossary/node.html), [argument of perihelion](https://ssd.jpl.nasa.gov/glossary/arg_peri.html), [true anomaly](https://ssd.jpl.nasa.gov/glossary/true_anomaly.html), [mean motion](https://ssd.jpl.nasa.gov/glossary/n.html) · [CSUN — the three anomalies](http://www.csun.edu/~hcmth017/master/node14.html) · [orbital-mechanics.space — Classical Orbital Elements](https://orbital-mechanics.space/classical-orbital-elements/classical-orbital-elements.html) · [René Schwarz Memo 1 (PDF)](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf)

### B.3 Kepler's Equation and Newton–Raphson Solution

$$M = E - e\sin E, \qquad E_{n+1} = E_n - \frac{E_n - e\sin E_n - M}{1 - e\cos E_n}, \quad E_0 = M$$

```
f(E)  = E - e*sin(E) - M
fp(E) = 1 - e*cos(E)
E = M                      # initial guess
repeat: dE = f(E)/fp(E); E = E - dE
until abs(dE) < tol        # e.g. 1e-12 rad
```

- Transcendental (no closed form); Newton–Raphson converges quadratically for elliptic orbits.

Sources: [René Schwarz Memo 1, eqs. 3–4](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf) · [JPL fltops ch. 9, eqs. 9.6.5/9.6.8](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/CelestialMechanics9-2%20body%20prob.pdf) · [ASU MAE462 Lecture 4](https://control.asu.edu/Classes/MAE462/462Lecture04.pdf) · [NASA TN — Optimized solution of Kepler's equation](https://ntrs.nasa.gov/api/citations/19720016564/downloads/19720016564.pdf)

### B.4 True Anomaly from Eccentric Anomaly

$$\tan\frac{\nu}{2} = \sqrt{\frac{1+e}{1-e}}\,\tan\frac{E}{2} \qquad\Longleftrightarrow\qquad \nu = 2\,\mathrm{atan2}\!\left(\sqrt{1+e}\,\sin\tfrac{E}{2},\ \sqrt{1-e}\,\cos\tfrac{E}{2}\right)$$

```
nu = 2 * atan( sqrt((1+e)/(1-e)) * tan(E/2) )                    # branch issues near E = +/-pi
nu = 2 * atan2( sqrt(1+e)*sin(E/2), sqrt(1-e)*cos(E/2) )         # robust, full range — use in code
```

Sources: [René Schwarz Memo 1, eqs. 5–6](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf) · [ASU MAE462 Lecture 4](https://control.asu.edu/Classes/MAE462/462Lecture04.pdf) · [Wikipedia — Eccentric anomaly](https://en.wikipedia.org/wiki/Eccentric_anomaly)

### B.5 Orbital Radius

$$r = a(1 - e\cos E) = \frac{a(1-e^2)}{1+e\cos\nu} = \frac{p}{1+e\cos\nu}, \qquad p = a(1-e^2) = h^2/\mu$$

```
r_from_E  = a * (1 - e*cos(E))
p         = a * (1 - e*e)          # semi-latus rectum = h^2/mu
r_from_nu = p / (1 + e*cos(nu))
```

Consistency: with `cos E = (e + cos nu)/(1 + e cos nu)`, `a(1 - e cos E) = a(1-e^2)/(1+e cos nu)` exactly.

Sources: [Wikipedia — Eccentric anomaly](https://en.wikipedia.org/wiki/Eccentric_anomaly) · [CU Boulder Intro Orbital Mechanics ch. 6](https://colorado.pressbooks.pub/introorbitalmechanics/chapter/chapter-6-keplers-prediction-problem/) · [Oregon State space systems notes](https://kyleniemeyer.github.io/space-systems-notes/orbital-mechanics/two-body-problems.html) · [JPL fltops ch. 9 (eq. 2.3.16)](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/CelestialMechanics9-2%20body%20prob.pdf) · [René Schwarz Memo 1, eq. 7](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf)

### B.6 Position and Velocity in the Perifocal (PQW) Frame

$$\vec r_{PQW} = r\begin{bmatrix}\cos\nu\\ \sin\nu\\ 0\end{bmatrix},\qquad \vec v_{PQW} = \frac{\mu}{h}\begin{bmatrix}-\sin\nu\\ e+\cos\nu\\ 0\end{bmatrix}$$

$$\vec r_{PQW} = \begin{bmatrix}a(\cos E - e)\\ a\sqrt{1-e^2}\sin E\\ 0\end{bmatrix},\qquad \vec v_{PQW} = \frac{\sqrt{\mu a}}{r}\begin{bmatrix}-\sin E\\ \sqrt{1-e^2}\cos E\\ 0\end{bmatrix}$$

```
# true-anomaly form
r     = p / (1 + e*cos(nu))
r_pqw = [ r*cos(nu), r*sin(nu), 0 ]
v_pqw = (mu/h) * [ -sin(nu), e + cos(nu), 0 ]
# eccentric-anomaly form (r = a*(1 - e*cos(E)))
r_pqw = [ a*(cos(E)-e), a*sqrt(1-e*e)*sin(E), 0 ]
v_pqw = (sqrt(mu*a)/r) * [ -sin(E), sqrt(1-e*e)*cos(E), 0 ]
```

- Basis: `P_hat` toward periapsis, `Q_hat` 90° ahead in-plane, `W_hat = P x Q` along `h_vec`; `h = |r x v| = sqrt(mu*p)` [m^2/s].

Sources: [René Schwarz Memo 1, eq. 8](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf) · [orbital-mechanics.space — Perifocal frame](https://orbital-mechanics.space/classical-orbital-elements/perifocal-frame.html) · [Wikipedia — Perifocal coordinate system](https://en.wikipedia.org/wiki/Perifocal_coordinate_system)

### B.7 Perifocal-to-Ecliptic Rotation Matrix

Convention (verified): passive 3-1-3 sequence; inertial-to-perifocal is `R_3(omega) R_1(i) R_3(Omega)`, so perifocal-to-inertial (what the simulator applies) is

$$\vec r_{IJK} = R_z(-\Omega)\,R_x(-i)\,R_z(-\omega)\;\vec r_{PQW}$$

$$R = \begin{bmatrix}
\cos\Omega\cos\omega - \sin\Omega\sin\omega\cos i & -\cos\Omega\sin\omega - \sin\Omega\cos\omega\cos i & \sin\Omega\sin i \\
\sin\Omega\cos\omega + \cos\Omega\sin\omega\cos i & -\sin\Omega\sin\omega + \cos\Omega\cos\omega\cos i & -\cos\Omega\sin i \\
\sin\omega\sin i & \cos\omega\sin i & \cos i
\end{bmatrix}$$

```
cO=cos(Omega); sO=sin(Omega); ci=cos(i); si=sin(i); cw=cos(omega); sw=sin(omega)
R11 =  cO*cw - sO*sw*ci;   R12 = -cO*sw - sO*cw*ci;   R13 =  sO*si
R21 =  sO*cw + cO*sw*ci;   R22 = -sO*sw + cO*cw*ci;   R23 = -cO*si
R31 =  sw*si;              R32 =  cw*si;              R33 =  ci
# r_ecl = R @ r_pqw ; same R applies to v_pqw. R is orthogonal: R^-1 = R^T.
```

- Column 3 of `R` is the orbit normal in inertial coordinates, `h_hat = (sin Omega sin i, -cos Omega sin i, cos i)` — consistent with `i = arccos(h_z/h)` and the node vector in B.8. Verified three ways: direct multiplication, ASU active-form `R_3(Omega)R_1(i)R_3(omega)`, and Schwarz's identical `R_z(-Omega)R_x(-i)R_z(-omega)` with fully expanded components.

Sources: [René Schwarz Memo 1, eqs. 9–10](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf) · [ASU MAE462 Lecture 7](https://control.asu.edu/Classes/MAE462/462Lecture07.pdf) · [Wikipedia — Perifocal coordinate system](https://en.wikipedia.org/wiki/Perifocal_coordinate_system)

### B.8 Inverse Problem: Osculating Elements from State Vector

Given `r_vec`, `v_vec`, `mu`:

$$\vec h = \vec r \times \vec v,\qquad \vec n = \hat K \times \vec h = (-h_y, h_x, 0),\qquad \vec e = \frac{\vec v \times \vec h}{\mu} - \frac{\vec r}{r}$$

$$\varepsilon = \frac{v^2}{2} - \frac{\mu}{r},\qquad a = -\frac{\mu}{2\varepsilon},\qquad e = |\vec e|,\qquad i = \arccos\frac{h_z}{h}$$

$$\Omega = \begin{cases}\arccos(n_x/n) & n_y \ge 0\\ 2\pi-\arccos(n_x/n) & n_y < 0\end{cases}\quad
\omega = \begin{cases}\arccos\frac{\vec n\cdot\vec e}{n e} & e_z \ge 0\\ 2\pi-\arccos\frac{\vec n\cdot\vec e}{n e} & e_z < 0\end{cases}\quad
\nu = \begin{cases}\arccos\frac{\vec e\cdot\vec r}{e r} & \vec r\cdot\vec v \ge 0\\ 2\pi-\arccos\frac{\vec e\cdot\vec r}{e r} & \vec r\cdot\vec v < 0\end{cases}$$

```
h_vec = cross(r_vec, v_vec);  h = norm(h_vec)
n_vec = cross([0,0,1], h_vec);  n = norm(n_vec)         # = (-h_y, h_x, 0)
e_vec = cross(v_vec, h_vec)/mu - r_vec/norm(r_vec);  e = norm(e_vec)
eps   = dot(v_vec,v_vec)/2 - mu/norm(r_vec)
a     = -mu / (2*eps)                                    # = 1/(2/r - v^2/mu)
i     = acos(h_vec[2]/h)                                 # [0, pi], unambiguous
Omega = acos(n_vec[0]/n);        if n_vec[1] < 0: Omega = 2*pi - Omega
omega = acos(dot(n_vec,e_vec)/(n*e));  if e_vec[2] < 0: omega = 2*pi - omega
nu    = acos(dot(e_vec,r_vec)/(e*norm(r_vec)));  if dot(r_vec,v_vec) < 0: nu = 2*pi - nu
```

- `e_vec` — eccentricity vector (toward periapsis, |e_vec| = e); Laplace–Runge–Lenz vector over `mu`. `eps` — specific orbital energy [m^2/s^2].
- **Degenerate cases** (per Vallado-style `rv2coe`): equatorial (`n ≈ 0`) — use true longitude (angle from x-axis to `r_vec`, sign from `r_y`); circular (`e ≈ 0`) — use argument of latitude (angle from `n_vec` to `r_vec`, sign from `r_z`).

Sources: [René Schwarz Memo 2 (PDF)](https://downloads.rene-schwarz.com/download/M002-Cartesian_State_Vectors_to_Keplerian_Orbit_Elements.pdf) · [orbital-mechanics.space — Elements and the state vector](https://orbital-mechanics.space/classical-orbital-elements/orbital-elements-and-the-state-vector.html) · [Vallado-style RV2COE.m implementation](https://github.com/dinkelk/astrodynamics/blob/master/astrodynamics/RV2COE.m)

### B.9 Vis-Viva Equation

$$v^2 = \mu\left(\frac{2}{r}-\frac{1}{a}\right)$$

```
v = sqrt( mu * (2.0/r - 1.0/a) )
```

- From `eps = v^2/2 - mu/r = -mu/(2a) = const`; `a>0` ellipse, `a->inf` parabola, `a<0` hyperbola.

Sources: [JPL fltops ch. 9, eq. 9.5.31](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/CelestialMechanics9-2%20body%20prob.pdf) · [Wikipedia — Vis-viva equation](https://en.wikipedia.org/wiki/Vis-viva_equation)

### B.10 Mean Motion

$$n = \sqrt{\frac{\mu}{a^3}} = \frac{2\pi}{T}$$

```
n = sqrt(mu / (a*a*a))          # rad/s;  M(t) = M0 + n*(t - t0)
```

Sources: [JPL SSD glossary — mean motion](https://ssd.jpl.nasa.gov/glossary/n.html) · [Wikipedia — Mean motion](https://en.wikipedia.org/wiki/Mean_motion) · [René Schwarz Memo 1, eq. 2](https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf)

### B.11 Orbital Period

$$T = 2\pi\sqrt{\frac{a^3}{\mu}}$$

```
T = 2.0*pi * sqrt(a*a*a / mu)
```

Sources: [JPL fltops ch. 9, eq. 9.2.2/9.5.23](https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/CelestialMechanics9-2%20body%20prob.pdf) · [Wikipedia — Mean motion](https://en.wikipedia.org/wiki/Mean_motion)

---

## C. General Relativity (Weak-Field)

### C.1 1PN Correction Acceleration (EIH Two-Body Reduction — what ephemerides use)

$$\vec{a}_{\rm GR} = \frac{GM}{c^{2}r^{3}}\left[\left(\frac{4GM}{r} - v^{2}\right)\vec{r} + 4\left(\vec{r}\cdot\vec{v}\right)\vec{v}\right]$$

added to the Newtonian term `a_N = -GM*r_vec/r^3`.

```
a_GR = (G*M/(c^2 * r^3)) * ( (4*G*M/r - v^2)*r_vec + 4*dot(r_vec, v_vec)*v_vec )
```

- `r_vec` — position of orbiting body relative to central mass [m], `r = |r_vec|`; `v_vec` — relative velocity [m/s], `v = |v_vec|`. Geometrized: `a_GR = (M/r^3)((4M/r - v^2) r_vec + 4 (r.v) v_vec)`.
- **Coefficient confirmation:** IERS Conventions (2010) Ch. 10, Eq. 10.12 gives the PPN Schwarzschild-term acceleration `Δr̈ = GM/(c²r³){[2(β+γ)GM/r − γ v²]r + 2(1+γ)(r·v)v}`; substituting GR values **β = γ = 1** yields exactly `(4GM/r − v²)r + 4(r·v)v`. Same form in the Explanatory Supplement to the Astronomical Almanac (p. 281, as reproduced by Project Pluto). This is the dominant term of the full N-body PPN/EIH law used in JPL DE430/DE440 (their Eq. 27, β=γ=1); full N-body treatment: Damour, Soffel & Xu, Phys. Rev. D 43–47 (1991–94).

Sources: [IERS Conventions 2010, Ch. 10 (Eq. 10.12)](https://iers-conventions.obspm.fr/content/chapter10/tn36_c10.pdf) · [Project Pluto — relativistic corrections (Expl. Suppl. p. 281)](https://www.projectpluto.com/relativi.htm) · [Folkner et al., DE430/DE431 (Eq. 27)](https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/de430_and_de431.pdf) · [Park et al. 2021, DE440/DE441 (Eq. 27)](https://ssd.jpl.nasa.gov/doc/Park.2021.AJ.DE440.pdf) · [Damour, Soffel & Xu I, PRD 43, 3273](https://link.aps.org/doi/10.1103/PhysRevD.43.3273)

### C.2 Perihelion Advance per Orbit

$$\Delta\omega = \frac{6\pi GM}{c^{2}a(1-e^{2})}$$

```
delta_omega = 6*pi*G*M / (c^2 * a * (1 - e^2))     # radians per orbit
```

- `a` — semi-major axis [m]; `e` — eccentricity. Geometrized: `6*pi*M/(a*(1-e^2))`.
- **Mercury:** GR contribution **42.98 arcsec/century** (precisely 42.9799″/cy, quoted as (42.980 ± 0.001)″/cy; total observed 574.10″/cy, of which 532.3035″/cy is Newtonian planetary perturbation). Independent recomputation with `a = 5.79090e10 m`, `e = 0.20563`, `T = 87.9691 d`, `GM_sun = 1.32712440018e20` gives 0.10353″/orbit × 415.20 orbits/cy ≈ 42.998″/cy — confirming the `6 pi` coefficient.

Sources: [Wikipedia — Tests of general relativity](https://en.wikipedia.org/wiki/Tests_of_general_relativity) · [Wikipedia — Schwarzschild geodesics](https://en.wikipedia.org/wiki/Schwarzschild_geodesics) · [Tipler Modern Physics companion — Perihelion of Mercury (43.0″/cy)](https://www.macmillanlearning.com/studentresources/college/physics/tiplermodernphysics6e/more_sections/more_chapter_2_1-perihelion_of_mercurys_orbit.pdf)

### C.3 Gravitational Time Dilation (Static Observer)

$$d\tau = \sqrt{1-\frac{2GM}{rc^{2}}}\;dt$$

```
dtau_dt = sqrt(1 - 2*G*M/(r*c^2))
```

- `dtau` — proper time of the static observer at areal radius `r` [s]; `dt` — coordinate time (static observer at infinity) [s]. Direct `g_tt` of the Schwarzschild metric; weak-field expansion `dtau/dt ≈ 1 + Phi/c^2`, `Phi = -GM/r` (GPS literature form).

Sources: [U. Alabama Schwarzschild solution notes](https://pages.physics.ua.edu/staff/fabi/phRel/SWSolution.pdf) · [Ashby, "Relativity in the GPS," Living Rev. Relativity 6, 1 (2003), Eq. 27](https://link.springer.com/article/10.12942/lrr-2003-1)

### C.4 Total Time Dilation, Circular Orbit

$$\frac{d\tau}{dt} = \sqrt{1-\frac{3GM}{rc^{2}}}$$

```
dtau_dt_orbit = sqrt(1 - 3*G*M/(r*c^2))
```

- **Why 3, not 2:** gravitational term contributes `2GM/(rc^2)` (from `g_tt`), and the kinematic term adds `v^2/c^2 = GM/(rc^2)` because circular Schwarzschild orbits satisfy exactly `v^2 = GM/r` (Kepler III holds exactly in these coordinates, D.7): `dτ²/dt² = (1 − 2GM/rc²) − GM/rc² = 1 − 3GM/rc²`. Linearized, this is Ashby Eq. 35's GPS clock-rate offset `3GM/(2ac^2)`.

Sources: [Ashby, Living Rev. Relativity 6, 1 (2003), Eqs. 27 & 35](https://link.springer.com/article/10.12942/lrr-2003-1) · [MathPages — orbital proper time in Schwarzschild geometry](https://www.mathpages.com/rr/s6-05/6-05.htm)

### C.5 Gravitational Redshift

$$\frac{\lambda_{o}}{\lambda_{e}} = \frac{\nu_{e}}{\nu_{o}} = \sqrt{\frac{1-\dfrac{2GM}{r_{o}c^{2}}}{1-\dfrac{2GM}{r_{e}c^{2}}}}$$

```
lambda_obs/lambda_emit = sqrt( (1 - 2*G*M/(r_obs*c^2)) / (1 - 2*G*M/(r_emit*c^2)) )
z = lambda_obs/lambda_emit - 1
```

- `r_e, r_o` — areal radii of emitter and observer [m]. With `r_s = 2GM/c^2`: `sqrt((1 - r_s/r_o)/(1 - r_s/r_e))`; linearized: `dnu/nu ≈ (GM/c^2)(1/r_e - 1/r_o)` (surface form `gh/c^2`).

Sources: [Wikipedia — Gravitational redshift](https://en.wikipedia.org/wiki/Gravitational_redshift) · [arXiv:2309.10499, Eq. 2.4–2.5](https://arxiv.org/pdf/2309.10499)

### C.6 Shapiro Delay

General (integral) form, PPN coefficient `(1+gamma)` with `gamma = 1`:

$$\Delta t = \frac{2GM}{c^{3}}\int \frac{dl}{r(l)}$$

Closed form, one-way between emitter (distance `r_E` from `M`) and receiver (`r_P`), separation `r_EP`, impact parameter `d`:

$$\Delta t = \frac{2GM}{c^{3}}\ln\!\left[\frac{r_{E}+r_{P}+r_{EP}}{r_{E}+r_{P}-r_{EP}}\right] \approx \frac{2GM}{c^{3}}\ln\!\left(\frac{4\,r_{E}r_{P}}{d^{2}}\right)$$

```
delta_t           = (2*G*M/c^3) * ln( (r_E + r_P + r_EP) / (r_E + r_P - r_EP) )
delta_t_approx    = (2*G*M/c^3) * ln( 4*r_E*r_P / d^2 )
delta_t_roundtrip = 2 * delta_t          # radar echo (original Shapiro test): coefficient 4GM/c^3
```

Sources: [Pössel, "The Shapiro time delay and the equivalence principle," arXiv:2001.00229 (Eqs. 12, 27)](https://arxiv.org/pdf/2001.00229) · [Wikipedia — Shapiro time delay](https://en.wikipedia.org/wiki/Shapiro_time_delay)

### C.7 Light Deflection

$$\alpha = \frac{4GM}{c^{2}b}$$

```
alpha = 4*G*M / (c^2 * b)      # radians
```

- `b` — impact parameter [m]. Geometrized: `alpha = 4M/b = 2 r_s/b`. Exactly **twice** the Newtonian/equivalence-principle-only value; grazing the Sun: 1.75″ (Newtonian: 0.87″).

Sources: [arXiv:2405.04529 — deflection of light, Eq. 17](https://arxiv.org/html/2405.04529v1) · [Wikipedia — Tests of general relativity](https://en.wikipedia.org/wiki/Tests_of_general_relativity)

### C.8 Einstein Ring Angular Radius

$$\theta_{E} = \sqrt{\frac{4GM}{c^{2}}\cdot\frac{D_{LS}}{D_{L}D_{S}}}$$

```
theta_E = sqrt( (4*G*M/c^2) * D_LS / (D_L * D_S) )     # radians
```

- `M` — lens mass [kg]; `D_L, D_S, D_LS` — angular diameter distances observer→lens, observer→source, lens→source [m]. From the point-mass lens equation with source offset zero.

Source: [LibreTexts — Lensing by Point Masses](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Big_Ideas_in_Cosmology_(Coble_et_al.)/12:_Gravitational_Lenses/12.02:_Lensing_by_Point_Masses)

---

## D. Black Holes

### Schwarzschild

#### D.1 Schwarzschild Metric Line Element

$$ds^2 = -\left(1-\frac{2GM}{c^2 r}\right)c^2\,dt^2 + \left(1-\frac{2GM}{c^2 r}\right)^{-1}dr^2 + r^2\left(d\theta^2+\sin^2\theta\,d\phi^2\right)$$

```
ds^2 = -(1 - 2GM/(c^2 r)) c^2 dt^2 + (1 - 2GM/(c^2 r))^(-1) dr^2 + r^2 (dtheta^2 + sin(theta)^2 dphi^2)
# with r_s = 2GM/c^2:  ds^2 = -(1 - r_s/r) c^2 dt^2 + dr^2/(1 - r_s/r) + r^2 dOmega^2
```

- `t` — coordinate time [s]; `r` — areal radius (sphere at fixed r has area `4 pi r^2`) [m]; `theta, phi` — angles [rad].

Sources: [Wikipedia — Schwarzschild metric](https://en.wikipedia.org/wiki/Schwarzschild_metric) · [Carroll GR lecture notes ch. 7](https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html) · [MIT OCW 8.033 Schwarzschild notes](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/5dae4a0c2d69b367951ddea5e17a2cea_schwarzschild.pdf)

#### D.2 Event Horizon

$$r_s = \frac{2GM}{c^2}$$

```
r_s = 2*G*M/c^2        # geometrized: r_s = 2M
```

Source: [Wikipedia — Schwarzschild metric](https://en.wikipedia.org/wiki/Schwarzschild_metric)

#### D.3 Photon Sphere

$$r_{\rm photon} = \frac{3GM}{c^2} = \tfrac{3}{2}r_s$$

```
r_photon = 3*G*M/c^2 = 1.5*r_s
```

- Unstable circular orbit for light, same radius for all photon angular momenta.

Sources: [Wikipedia — Photon sphere](https://en.wikipedia.org/wiki/Photon_sphere) · [Carroll notes, eq. 7.51](https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html)

#### D.4 ISCO (Massive Particles)

$$r_{\rm ISCO} = \frac{6GM}{c^2} = 3r_s$$

```
r_ISCO = 6*G*M/c^2
```

Sources: [Wikipedia — ISCO](https://en.wikipedia.org/wiki/Innermost_stable_circular_orbit) · [Carroll notes, eq. 7.55](https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html) · [MIT OCW 8.033](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/5dae4a0c2d69b367951ddea5e17a2cea_schwarzschild.pdf) · [UMD astr498 lecture 10](https://pages.astro.umd.edu/~mcmiller/teaching/astr498/lecture10.pdf) · [Tübingen GTR notes](https://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/GTR_files/GTR2018_3b.pdf)

#### D.5 Effective Potential (Massive Particle)

**Form A** — additive potential, per unit mass, with equation of motion `(1/2)(dr/dtau)^2 + V_eff(r) = (c^2/2)[(E/mc^2)^2 - 1]`:

$$V_{\rm eff}(r) = -\frac{GM}{r} + \frac{L^2}{2r^2} - \frac{GML^2}{c^2 r^3}$$

**Form B** — energy-normalized fully relativistic form (algebraically identical: `(E/mc^2)^2 = 1 + 2V_eff/c^2`):

$$\left(\frac{E}{mc^2}\right)^2 = \left(1-\frac{2GM}{c^2 r}\right)\left(1+\frac{L^2}{c^2 r^2}\right)$$

```
V_eff(r) = -G*M/r + L^2/(2*r^2) - G*M*L^2/(c^2 * r^3)
(E/(m*c^2))^2 = (1 - 2*G*M/(c^2*r)) * (1 + L^2/(c^2*r^2))
(dr/dtau)^2   = c^2*(E/(m*c^2))^2 - c^2*(1 - 2*G*M/(c^2*r))*(1 + L^2/(c^2*r^2))
```

- `E` — conserved orbital energy [J]; `L` — specific angular momentum (per unit rest mass) [m^2/s]. The `-GML^2/(c^2 r^3)` term is the GR correction absent in Newtonian mechanics — it destabilizes circular orbits below `6GM/c^2` and permits plunge orbits. Geometrized: `V_eff = -M/r + L^2/(2r^2) - M L^2/r^3`.

Sources: [Carroll notes, eq. 7.48 (Form A verbatim)](https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html) · [Wikipedia — Schwarzschild geodesics](https://en.wikipedia.org/wiki/Schwarzschild_geodesics) · [MIT OCW 8.033 (Form B verbatim)](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/5dae4a0c2d69b367951ddea5e17a2cea_schwarzschild.pdf) · [LibreTexts / Skidmore GR — Effective Potential](https://phys.libretexts.org/Courses/Skidmore_College/Introduction_to_General_Relativity/03:_Schwarzschild_Orbits/3.03:_Effective_Potential) · [Grokipedia — ISCO](https://grokipedia.com/page/Innermost_stable_circular_orbit)

#### D.6 Specific Energy and Angular Momentum on Circular Orbits (and at ISCO)

$$\tilde{E}(r) = \frac{E}{mc^2} = \frac{1-\dfrac{2GM}{c^2 r}}{\sqrt{1-\dfrac{3GM}{c^2 r}}}, \qquad \tilde{L}(r) = \frac{L}{mc} = \frac{\sqrt{GMr/c^2}}{\sqrt{1-\dfrac{3GM}{c^2 r}}}$$

$$\frac{E}{mc^2}\bigg|_{\rm ISCO} = \frac{2\sqrt{2}}{3} \approx 0.9428, \qquad \frac{L}{mc}\bigg|_{\rm ISCO} = 2\sqrt{3}\,\frac{GM}{c^2}$$

```
E_tilde(r) = (1 - 2*G*M/(c^2*r)) / sqrt(1 - 3*G*M/(c^2*r))        # dimensionless
L_tilde(r) = sqrt(G*M*r/c^2) / sqrt(1 - 3*G*M/(c^2*r))            # L/(m*c), units of length
E_ISCO = 2*sqrt(2)/3 ~= 0.94281 ;  L_ISCO = 2*sqrt(3)*G*M/c^2 ~= 3.4641*G*M/c^2
```

- Binding energy from rest at infinity to ISCO: `1 - 2*sqrt(2)/3 ≈ 5.72%` of rest mass — the standard Schwarzschild thin-disk radiative efficiency.

Sources: [UMD astr498 lecture 10](https://pages.astro.umd.edu/~mcmiller/teaching/astr498/lecture10.pdf) · [Tübingen GTR notes, eqs. 26–27](https://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/GTR_files/GTR2018_3b.pdf) · [Carroll notes](https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html) · [MIT OCW 8.033](https://ocw.mit.edu/courses/8-033-relativity-fall-2006/5dae4a0c2d69b367951ddea5e17a2cea_schwarzschild.pdf) · [Grokipedia — ISCO](https://grokipedia.com/page/Innermost_stable_circular_orbit)

#### D.7 Circular-Orbit Angular Velocity

$$\Omega(r) = \frac{d\phi}{dt} = \sqrt{\frac{GM}{r^3}}$$

```
Omega(r) = sqrt(G*M / r^3)      # rad/s in coordinate time
```

- **Exactly the Newtonian/Keplerian form in Schwarzschild coordinate time** — a well-known coincidence (UMD notes: "By a lovely coincidence, in Schwarzschild coordinates the angular velocity observed at infinity is exactly the same as it is in Newtonian physics"). Distinct from the locally measured shell-observer speed `v_shell = sqrt((GM/r)/(1 - 2GM/(c^2 r)))`.

Sources: [UMD astr498 lecture 10 (verbatim quote)](https://pages.astro.umd.edu/~mcmiller/teaching/astr498/lecture10.pdf) · [Tübingen GTR notes, eq. 30](https://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/GTR_files/GTR2018_3b.pdf)

#### D.8 Radial Infall Proper Time

Drop from rest at `r0` (Newtonian-form energy equation holds exactly in proper time):

$$\left(\frac{dr}{d\tau}\right)^2 = 2GM\left(\frac{1}{r}-\frac{1}{r_0}\right)$$

From rest at infinity, between radii `r_i > r_f`:

$$\Delta\tau = \frac{2}{3\sqrt{2GM}}\left(r_i^{3/2}-r_f^{3/2}\right)$$

Finite `r0` — cycloid parametric solution (MTW/Chandrasekhar standard result), and total proper time to the singularity:

$$r(\eta) = \frac{r_0}{2}(1+\cos\eta),\quad \tau(\eta) = \sqrt{\frac{r_0^3}{8GM}}(\eta+\sin\eta),\quad \eta\in[0,\pi];\qquad \tau_{\rm total} = \frac{\pi r_0^{3/2}}{2\sqrt{2GM}}$$

```
(dr/dtau)^2 = 2*G*M*(1/r - 1/r0)
dtau_from_inf = (2/(3*sqrt(2*G*M))) * (r_i^1.5 - r_f^1.5)
r(eta)   = (r0/2)*(1 + cos(eta));   tau(eta) = sqrt(r0^3/(8*G*M)) * (eta + sin(eta))
tau_total = pi * r0^1.5 / (2*sqrt(2*G*M))
```

Sources: [Tübingen GTR notes, eq. 22](https://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/GTR_files/GTR2018_3b.pdf) · [Physics Forums — free fall to the singularity (τ = πr0^{3/2}/(2√(2M)))](https://www.physicsforums.com/threads/free-fall-from-rest-at-a-particular-radius-to-the-central-singularity.1079546/)

### Kerr

#### D.9 Outer and Inner Horizons

$$r_\pm = \frac{GM}{c^2}\left(1\pm\sqrt{1-a_*^2}\right), \qquad a = \frac{J}{Mc}, \qquad a_* = \frac{ac^2}{GM} = \frac{Jc}{GM^2}$$

```
a      = J/(M*c)                                # Kerr parameter [m]
a_star = a*c^2/(G*M) = J*c/(G*M^2)              # dimensionless spin, |a_star| <= 1
r_plus  = (G*M/c^2)*(1 + sqrt(1 - a_star^2))    # event horizon
r_minus = (G*M/c^2)*(1 - sqrt(1 - a_star^2))    # inner (Cauchy) horizon
```

- `J` — BH angular momentum [kg m^2/s]; `a* = 1` is extremal. Geometrized: `r_± = M ± sqrt(M^2 - a^2)`, with `Delta(r) = r^2 - 2Mr + a^2 = (r - r_+)(r - r_-)`.

Sources: [Roma1 INFN GR notes ch. 21 (eqs. 21.55–56)](https://www.roma1.infn.it/teongrav/onde19_20/kerr.pdf) · [UNCW Kerr metric notes](https://people.uncw.edu/hermanr/BlackHoles/Kerr_Metric_II.pdf) · [E. Taylor, "The Spinning Black Hole"](https://www.eftaylor.com/pub/SpinNEW.pdf)

#### D.10 Ergosphere (Static Limit)

$$r_{\rm ergo}(\theta) = \frac{GM}{c^2}\left(1+\sqrt{1-a_*^2\cos^2\theta}\right)$$

```
r_ergo(theta) = (G*M/c^2) * (1 + sqrt(1 - a_star^2 * cos(theta)^2))
```

- Ergoregion: `r_+ < r < r_ergo(theta)` — no static observers (mandatory co-rotation), escape still possible (Penrose process). Poles: `r_ergo = r_+`; equator: `r_ergo = 2GM/c^2` for any spin.

Sources: [Roma1 INFN GR notes, eqs. 21.65/21.69](https://www.roma1.infn.it/teongrav/onde19_20/kerr.pdf) · [UNCW Kerr notes (g_tt = 0 surface)](https://people.uncw.edu/hermanr/BlackHoles/Kerr_Metric_II.pdf) · [Wikipedia — Kerr metric](https://en.wikipedia.org/wiki/Kerr_metric)

#### D.11 Kerr ISCO — Bardeen–Press–Teukolsky (1972) Z1/Z2 Formulas

$$Z_1 = 1+(1-a_*^2)^{1/3}\left[(1+a_*)^{1/3}+(1-a_*)^{1/3}\right], \qquad Z_2 = \sqrt{3a_*^2+Z_1^2}$$

$$\frac{r_{\rm ISCO}}{GM/c^2} = 3+Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)} \qquad \begin{cases}-&\text{prograde } (L>0)\\ +&\text{retrograde } (L<0)\end{cases}$$

```
Z1 = 1 + (1 - a_star^2)^(1/3) * ( (1 + a_star)^(1/3) + (1 - a_star)^(1/3) )
Z2 = sqrt(3*a_star^2 + Z1^2)
r_ISCO = (G*M/c^2) * ( 3 + Z2 - sign * sqrt((3 - Z1)*(3 + Z1 + 2*Z2)) )
# sign = +1 prograde (minus branch), sign = -1 retrograde (plus branch)
```

- **Sanity checks:** `a*=0` → `Z1=Z2=3`, `r_ISCO = 6GM/c^2` (Schwarzschild ✓); `a*=1` → prograde `GM/c^2`, retrograde `9GM/c^2` (extremal limits ✓).
- **Primary source verified verbatim** (BPT 1972 Eq. 2.21, with sign convention quote: "the upper sign refers to direct orbits (corotating, L > 0) ... lower sign to retrograde").

Sources: [Bardeen, Press & Teukolsky 1972, ApJ 178, 347 — full-text PDF (Eq. 2.21)](https://the-center-of-gravity.com/documents/86/Bardeen-Press-Teukolsky_Rotating-Black-Holes-Locally-Nonrotating-Frames.pdf) · [Wikipedia — ISCO (cites BPT)](https://en.wikipedia.org/wiki/Innermost_stable_circular_orbit) · [Grokipedia — ISCO](https://grokipedia.com/page/Innermost_stable_circular_orbit)

#### D.12 Frame-Dragging Angular Velocity (ZAMO / Lense–Thirring)

Angular velocity of a zero-angular-momentum observer — the rate at which local inertial frames are dragged:

$$\omega(r,\theta) = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{2Mar}{(r^2+a^2)^2-a^2\Delta\sin^2\theta}, \qquad \Delta = r^2-2Mr+a^2 \quad (G=c=1)$$

$$\omega \xrightarrow{r\to\infty} \frac{2GJ}{c^2 r^3} \quad\text{(weak-field Lense–Thirring rate)}$$

```
# geometrized (G=c=1; M, a, r in length units):
Delta = r^2 - 2*M*r + a^2
omega = 2*M*a*r / ( (r^2 + a^2)^2 - a^2 * Delta * sin(theta)^2 )
# SI restoration (M -> G*M/c^2 inside Delta too; multiply by c for rad/s):
Delta_SI = r^2 - 2*G*M*r/c^2 + a^2
omega_SI = 2*G*M*a*r / ( c * ( (r^2 + a^2)^2 - a^2 * Delta_SI * sin(theta)^2 ) )     # rad/s
# weak field: omega ~= 2*G*J/(c^2 * r^3)
```

- Self-consistency: expanding at large r gives `2Ma/r^3` (geometrized); with `a = J/(Mc)`, `M -> GM/c^2`, this is `2GJ/(c^2 r^3)` — the standard Lense–Thirring result (MTW).

Sources: [Roma1 INFN GR notes, eqs. 21.24/21.27](https://www.roma1.infn.it/teongrav/onde19_20/kerr.pdf) · [UNCW Kerr notes (g_tφ, g_φφ components)](https://people.uncw.edu/hermanr/BlackHoles/Kerr_Metric_II.pdf) · [Wikipedia — Lense–Thirring precession](https://en.wikipedia.org/wiki/Lense%E2%80%93Thirring_precession)

---

## E. Quantum / Thermodynamic

### E.1 Hawking Temperature

$$T_H = \frac{\hbar c^3}{8\pi G M k_B}$$

```
T_H = (hbar * c^3) / (8 * pi * G * M * k_B)
# numerically: T_H[K] ~= 1.227e23 / M[kg]  ~=  6.17e-8 K * (M_sun / M)
```

- `T_H` [K]; `M` [kg]; constants as in the header. General form `T_H = hbar*kappa/(2*pi*k_B*c)` with surface gravity `kappa = c^4/(4GM)` for Schwarzschild. Matches Hawking (1975): T ≈ 10^26 (M/g)^-1 K.

Sources: [Hawking 1975, Comm. Math. Phys. 43, 199 — full text PDF](https://astrofrelat.fcaglp.unlp.edu.ar/agujeros_negros/media/Papers/Hawking_1975-Particle_creation_by_black_holes.pdf) · [Springer DOI 10.1007/BF02345020](https://link.springer.com/article/10.1007/BF02345020) · [Jacobson, Black Hole Thermodynamics lectures (UMD)](https://www.physics.umd.edu/grt/taj/776b/lectures.pdf) · [Wikipedia — Hawking radiation](https://en.wikipedia.org/wiki/Hawking_radiation)

### E.2 Evaporation Lifetime

$$t_{\rm ev} = \frac{5120\,\pi\, G^2 M^3}{\hbar c^4}$$

```
t_ev = 5120 * pi * G^2 * M^3 / (hbar * c^4)
# numerically: t_ev[s] ~= 8.41e-17 * (M[kg])^3 ;  M = M_sun -> ~2.1e67 yr
```

- Derivation: Stefan–Boltzmann horizon emission `P = sigma*A*T_H^4` with `A = 16 pi G^2 M^2/c^4` gives `dM/dt = -hbar c^4/(15360 pi G^2 M^2)`; integrating `M^2 dM` from `M` to 0 yields exactly `5120 pi G^2 M^3/(hbar c^4)`.
- Caveat: `5120 pi` is the idealized photons-only blackbody coefficient (the standard reference formula); greybody factors and multi-species emission (Page 1976) rescale it by O(1).

Sources: [Hawking radiation — archived Wikipedia (formula + numeric coefficient 8.407e-17 s/kg^3)](https://dlab.epfl.ch/wikispeedia/wpcd/wp/h/Hawking_radiation.htm) · [Wikipedia — Hawking radiation (10^67 yr solar-mass check)](https://en.wikipedia.org/wiki/Hawking_radiation) · [iCalculator BH temperature/evaporation](https://physics.icalculator.com/black-hole-temperature-calculator.html) · [V. Toth — Hawking radiation calculator](https://www.vttoth.com/CMS/physics-notes/311-hawking-radiation-calculator) · [LibreTexts — Quantum effects near black holes](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Big_Ideas_in_Cosmology_(Coble_et_al.)/11:_Black_Holes/11.03:_Quantum_Effects_Near_Black_Holes)

### E.3 Bekenstein–Hawking Entropy

$$S_{BH} = \frac{k_B c^3 A}{4 G \hbar}, \qquad A = 4\pi r_s^2 = \frac{16\pi G^2 M^2}{c^4} \qquad\Longrightarrow\qquad S_{BH} = \frac{4\pi k_B G M^2}{\hbar c}$$

```
r_s  = 2*G*M/c^2
A    = 4*pi*r_s^2                            # = 16*pi*G^2*M^2/c^4
S_BH = (k_B * c^3 * A) / (4 * G * hbar)      # area form [J/K]
S_BH = 4*pi*k_B*G*M^2 / (hbar*c)             # mass form (substitution)
```

- Bekenstein (1973) proposed S ∝ A (different prefactor); Hawking (1975) fixed the exact 1/4 coefficient (`S = c^3 A/(4 G hbar)`, k_B = 1 units).

Sources: [Jacobson lectures (S = A/4ħG)](https://www.physics.umd.edu/grt/taj/776b/lectures.pdf) · [nuclear-power.com — Bekenstein–Hawking entropy](https://www.nuclear-power.com/bekenstein-hawking-entropy/) · [Bekenstein 1973, PRD 7, 2333](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.7.2333) · [Hawking 1975](https://link.springer.com/article/10.1007/BF02345020) · [arXiv:2507.03778 (quotes both original equations)](https://arxiv.org/html/2507.03778v1)

### E.4 Numerical Reference Values

| Quantity | Value | Note |
|---|---|---|
| `r_s` of the Sun | **2.953 km** (2953.2 m via `GM_sun = 1.32712440018e20`; 2954 m via CODATA G × 1.989e30 kg) | ≈ 2.95 km ✓ |
| Hawking T, 1 `M_sun` BH | **6.17e-8 K** (`1.227e23 / 1.989e30`) | ≈ 6.2e-8 K ✓; LibreTexts quotes 6.1e-8 K |
| Sagittarius A* mass | **(4.297 ± 0.012) × 10^6 M_sun = 8.547e36 kg** | GRAVITY Collaboration (Abuter et al.) 2022, A&A 657, L12 (±0.25%; syst. ≈ 0.040e6 M_sun; R0 = 8277 ± 9 pc) |
| `r_s` of Sgr A* | **1.269e10 m ≈ 1.27e7 km ≈ 0.0849 AU** | Computed from 2GM/c²; Wikipedia independently quotes 0.08 AU / 12 million km ✓ |

```
r_s_sun   = 2*GM_sun/c^2                     = 2.953e3 m
T_H_sun   = 1.227e23 / 1.989e30              = 6.17e-8 K
M_sgrA    = 4.297e6 * 1.989e30               = 8.547e36 kg
r_s_sgrA  = 2*G*M_sgrA/c^2                   = 1.269e10 m
```

Sources: [Wikipedia — Schwarzschild radius (table)](https://en.wikipedia.org/wiki/Schwarzschild_radius) · [IAU 2015 Resolution B3 nominal constants](https://www.astronomy.ohio-state.edu/pogge.1/Ast2292/Docs/NominalConstants.pdf) · [LibreTexts — Quantum effects near black holes](https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Big_Ideas_in_Cosmology_(Coble_et_al.)/11:_Black_Holes/11.03:_Quantum_Effects_Near_Black_Holes) · [GRAVITY Collaboration 2022, A&A 657, L12 (full text)](https://www.aanda.org/articles/aa/full_html/2022/01/aa42465-21/aa42465-21.html) · [DOI 10.1051/0004-6361/202142465](https://doi.org/10.1051/0004-6361/202142465) · [ADS 2022A&A...657L..12G](https://ui.adsabs.harvard.edu/abs/2022A%26A...657L..12G) · [Wikipedia — Sagittarius A*](https://en.wikipedia.org/wiki/Sagittarius_A*)

---

## F. Relativistic Visuals

### F.1 Relativistic Doppler Factor δ

$$\delta = \frac{1}{\gamma\left(1-\beta\cos\theta\right)}, \qquad \nu_{\rm obs} = \delta\,\nu_{\rm emit}, \qquad \gamma = \frac{1}{\sqrt{1-\beta^2}}$$

```
gamma  = 1 / sqrt(1 - beta^2)
delta  = 1 / ( gamma * (1 - beta*cos(theta)) )
nu_obs = delta * nu_emit
```

- `beta = v/c` — emitter (disk material) speed fraction; `theta` — angle between emitter velocity and line of sight, in the observer frame. `theta=0` approaching → `delta>1` blueshift; `theta=90°` → transverse redshift `delta = 1/gamma < 1`; `theta=180°` receding → `delta<1`.

Sources: [Urry & Padovani 1995, PASP 107, 803, Appendix A (Eq. A1)](https://ned.ipac.caltech.edu/level5/Urry1/UrryP_appena.html) · [Bicknell, ANU High Energy Astrophysics Lecture 8 (Eq. 14)](https://www.mso.anu.edu.au/~geoff/HEA/8_Relativistic_Effects.pdf) · [Rybicki & Lightman ch. 4 (hosted PDF)](https://moodle2.units.it/pluginfile.php/470593/mod_resource/content/0/RL_Cap4.pdf) · [Wikipedia — Relativistic Doppler effect](https://en.wikipedia.org/wiki/Relativistic_Doppler_effect) · [Wikipedia — Relativistic beaming](https://en.wikipedia.org/wiki/Relativistic_beaming)

### F.2 Combined Gravitational + Kinematic Redshift for Disk Material (g-factor)

$$g \equiv \frac{\nu_{\rm obs}}{\nu_{\rm emit}} = \frac{1}{1+z} = \delta\,\sqrt{1-\frac{r_s}{r}} = \frac{\sqrt{1-\dfrac{2GM}{rc^{2}}}}{\gamma(1-\beta\cos\theta)} \quad\text{(Schwarzschild)}$$

```
r_s = 2*G*M/c^2
g   = delta * sqrt(1 - r_s/r)     # total observed/emitted frequency ratio for a disk element
```

- The standard decomposition used by disk-imaging codes: gravitational time-dilation factor (static observer at `r` vs infinity) times the local orbital-motion Doppler factor. For Kerr, frame dragging couples the effects — use the full metric (lapse + ZAMO angular velocity); the definition `g = nu_obs/nu_emit = 1/(1+z)` is general and reduces to the above for `a* = 0`.

Sources: [Dovčiak, Karas & Yaqoob 2004, A&A 413, 861 (Eqs. 10, 13)](https://www.aanda.org/articles/aa/pdf/2004/03/aah4692.pdf) · [Wikipedia — Gravitational redshift](https://en.wikipedia.org/wiki/Gravitational_redshift) · [Wikipedia — Relativistic beaming](https://en.wikipedia.org/wiki/Relativistic_beaming)

### F.3 Intensity Beaming: δ³ (Specific) vs δ⁴ (Bolometric) — CONFIRMED

$$I_{\nu,\rm obs}(\nu_{\rm obs}) = \delta^{3}\, I_{\nu,\rm emit}(\nu_{\rm emit}) \qquad\text{(specific / monochromatic intensity)}$$

$$I_{\rm bol,obs} = \delta^{4}\, I_{\rm bol,emit} \qquad\text{(bolometric / frequency-integrated)}$$

```
I_nu_obs  = delta^3 * I_nu_emit       # per unit frequency  [W m^-2 Hz^-1 sr^-1]
I_bol_obs = delta^4 * I_bol_emit      # frequency-integrated [W m^-2 sr^-1]
# power-law spectrum I_nu ~ nu^(-alpha): flux density at fixed nu_obs ~ delta^(3+alpha);
# integrated flux ~ delta^(4+alpha)
```

- **Why:** `I_nu / nu^3` is Lorentz-invariant (photon phase-space density). With `nu_obs = delta*nu_emit`, the per-frequency intensity picks up **three** powers of delta (one from time compression, two from solid-angle aberration); integrating over frequency adds one more power from the Jacobian `dnu_obs = delta*dnu_emit`, giving **four** for bolometric. **For black-hole disk rendering, replace `delta` by the full g-factor of F.2** — exactly what the disk literature does (Dovčiak et al.: `I_obs,nu = g^3 I_em,nu`, integrated flux ∝ `g^4`).

Sources: [Urry & Padovani 1995, Appendix B (Eqs. B3–B5)](https://ned.ipac.caltech.edu/level5/Urry1/UrryP_appenb.html) · [Bicknell, ANU HEA Lecture 8 (Eqs. 99–100, 116)](https://www.mso.anu.edu.au/~geoff/HEA/8_Relativistic_Effects.pdf) · [Rybicki & Lightman ch. 4 (Eq. 4.110)](https://moodle2.units.it/pluginfile.php/470593/mod_resource/content/0/RL_Cap4.pdf) · [Dovčiak, Karas & Yaqoob 2004 (Eqs. 38, 41)](https://www.aanda.org/articles/aa/pdf/2004/03/aah4692.pdf) · [Wikipedia — Relativistic beaming](https://en.wikipedia.org/wiki/Relativistic_beaming)

### F.4 Shakura–Sunyaev Thin-Disk Temperature Profile

$$T(r) = \left\{ \frac{3\,G M \dot{M}}{8\pi\,\sigma_{\rm SB}\,r^{3}} \left[ 1 - \sqrt{\frac{r_{\rm in}}{r}}\right] \right\}^{1/4} \qquad\xrightarrow{r \gg r_{\rm in}}\qquad T(r) \approx \left(\frac{3GM\dot M}{8\pi\sigma_{\rm SB}r^{3}}\right)^{1/4} \propto r^{-3/4}$$

```
T(r) = ( (3*G*M*Mdot) / (8*pi*sigma_SB*r^3) * (1 - sqrt(r_in/r)) )^0.25
# far from inner edge: T ~ r^(-3/4)
```

- `M` — central mass [kg]; `Mdot` — accretion rate [kg/s]; `sigma_SB` — Stefan–Boltzmann constant [W m^-2 K^-4]; `r_in` — inner-edge radius [m] (= r_ISCO, F.5). From steady-state viscous dissipation `D(r) = (3GM*Mdot/(8 pi r^3))(1 - sqrt(r_in/r))` with the zero-torque inner boundary condition and two-sided radiation `2 sigma T^4 = D(r)`. Note `T(r_in) = 0` at the edge; peak T at `r = (49/36) r_in`.

Sources: [Armitage, "Lecture notes on accretion disk physics," arXiv:2201.07262 (Eqs. 123, 130)](https://arxiv.org/pdf/2201.07262) · [UMD astr498 lecture 12 (Eqs. 3, 9)](https://pages.astro.umd.edu/~mcmiller/teaching/astr498/lecture12.pdf) · Shakura & Sunyaev 1973, A&A 24, 337 — [ADS record](https://ui.adsabs.harvard.edu/abs/1973A%26A....24..337S/abstract) · [Frank, King & Raine, *Accretion Power in Astrophysics*, §5.3/§5.6 (full text)](https://www3.mpifr-bonn.mpg.de/staff/mmassi/frankkingraine.pdf) · [Wikipedia — Accretion disk](https://en.wikipedia.org/wiki/Accretion_disk)

### F.5 Disk Inner Edge at the ISCO

$$r_{\rm in} = r_{\rm ISCO}; \qquad r_{\rm ISCO}^{\rm Schw} = \frac{6GM}{c^2}; \qquad \frac{r_{\rm ISCO}^{\rm Kerr}}{GM/c^2} = 3+Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)}$$

```
r_in = r_ISCO
# Schwarzschild: r_ISCO = 6*G*M/c^2
# Kerr: use Z1/Z2 formulas of D.11 (minus = prograde, plus = retrograde)
# limits: a*=0 -> 6GM/c^2 ; a*=1 prograde -> GM/c^2 ; a*=1 retrograde -> 9GM/c^2
```

- Standard modeling choice for a non-truncated thin disk: circular orbits cease below the ISCO, justifying the zero-torque boundary there (Armitage notes the identification is physically motivated though "not entirely rigorous"). Radiative efficiency: ≈ 5.7% (Schwarzschild, cf. D.6) up to ≈ 42% (extremal prograde Kerr).

Sources: [Wikipedia — ISCO ("marks the inner edge of the disk")](https://en.wikipedia.org/wiki/Innermost_stable_circular_orbit) · [Armitage, arXiv:2201.07262, p. 15](https://arxiv.org/pdf/2201.07262) · [UMD astr498 lecture 12](https://pages.astro.umd.edu/~mcmiller/teaching/astr498/lecture12.pdf) · [Bardeen, Press & Teukolsky 1972, ApJ 178, 347 (Eq. 2.21)](https://the-center-of-gravity.com/documents/86/Bardeen-Press-Teukolsky_Rotating-Black-Holes-Locally-Nonrotating-Frames.pdf)

---

## Consolidated Source List

**NASA / JPL / agency**
- JPL Solar System Dynamics — Astrodynamic Parameters: https://ssd.jpl.nasa.gov/astro_par.html
- JPL SSD Glossary (semi-major axis, eccentricity, inclination, node, argument of perihelion, true anomaly, mean motion): https://ssd.jpl.nasa.gov/glossary/
- JPL fltops — Celestial Mechanics ch. 9, The Two-Body Problem: https://spsweb.fltops.jpl.nasa.gov/portaldataops/mpg/MPG_Docs/Source%20Docs/CelestialMechanics9-2%20body%20prob.pdf
- NASA Science — Orbits and Kepler's Laws: https://science.nasa.gov/solar-system/orbits-and-keplers-laws/
- NASA Basics of Spaceflight glossary: https://science.nasa.gov/learn/basics-of-space-flight/glossary/
- NASA TN D-6883 — Optimized solution of Kepler's equation: https://ntrs.nasa.gov/api/citations/19720016564/downloads/19720016564.pdf
- Folkner et al., JPL DE430/DE431 ephemeris paper: https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/de430_and_de431.pdf
- Park et al. 2021, JPL DE440/DE441 ephemeris paper: https://ssd.jpl.nasa.gov/doc/Park.2021.AJ.DE440.pdf

**IERS conventions**
- IERS Conventions (2010), Chapter 10 (Eq. 10.12, PPN Schwarzschild acceleration): https://iers-conventions.obspm.fr/content/chapter10/tn36_c10.pdf

**Primary literature**
- Bardeen, Press & Teukolsky 1972, ApJ 178, 347 (Kerr ISCO Z1/Z2): https://the-center-of-gravity.com/documents/86/Bardeen-Press-Teukolsky_Rotating-Black-Holes-Locally-Nonrotating-Frames.pdf
- Hawking 1975, Comm. Math. Phys. 43, 199: https://astrofrelat.fcaglp.unlp.edu.ar/agujeros_negros/media/Papers/Hawking_1975-Particle_creation_by_black_holes.pdf · https://link.springer.com/article/10.1007/BF02345020
- Bekenstein 1973, PRD 7, 2333: https://journals.aps.org/prd/abstract/10.1103/PhysRevD.7.2333
- GRAVITY Collaboration 2022, A&A 657, L12 (Sgr A* mass): https://www.aanda.org/articles/aa/full_html/2022/01/aa42465-21/aa42465-21.html · https://doi.org/10.1051/0004-6361/202142465 · https://ui.adsabs.harvard.edu/abs/2022A%26A...657L..12G
- Shakura & Sunyaev 1973, A&A 24, 337: https://ui.adsabs.harvard.edu/abs/1973A%26A....24..337S/abstract
- Urry & Padovani 1995, PASP 107, 803 (App. A/B): https://ned.ipac.caltech.edu/level5/Urry1/UrryP_appena.html · https://ned.ipac.caltech.edu/level5/Urry1/UrryP_appenb.html
- Dovčiak, Karas & Yaqoob 2004, A&A 413, 861: https://www.aanda.org/articles/aa/pdf/2004/03/aah4692.pdf
- Damour, Soffel & Xu 1991, PRD 43, 3273 (N-body PN celestial mechanics): https://link.aps.org/doi/10.1103/PhysRevD.43.3273
- Ashby 2003, Living Rev. Relativity 6, 1 (GPS relativity): https://link.springer.com/article/10.12942/lrr-2003-1
- Armitage, accretion disk lecture notes, arXiv:2201.07262: https://arxiv.org/pdf/2201.07262
- Pössel, Shapiro delay, arXiv:2001.00229: https://arxiv.org/pdf/2001.00229
- Light deflection pedagogical derivation, arXiv:2405.04529: https://arxiv.org/html/2405.04529v1
- Gravitational redshift revisited, arXiv:2309.10499: https://arxiv.org/pdf/2309.10499
- Black hole thermodynamics review, arXiv:2507.03778: https://arxiv.org/html/2507.03778v1

**Textbooks / monographs online**
- Sean Carroll, Lecture Notes on GR, ch. 7 (Caltech/NED mirror): https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll7.html
- Frank, King & Raine, *Accretion Power in Astrophysics* (full text): https://www3.mpifr-bonn.mpg.de/staff/mmassi/frankkingraine.pdf
- Rybicki & Lightman, *Radiative Processes*, ch. 4 (hosted): https://moodle2.units.it/pluginfile.php/470593/mod_resource/content/0/RL_Cap4.pdf
- Edwin Taylor, "The Spinning Black Hole" (Taylor & Wheeler companion): https://www.eftaylor.com/pub/SpinNEW.pdf
- Explanatory Supplement to the Astronomical Almanac p. 281, via Project Pluto: https://www.projectpluto.com/relativi.htm
- Klioner, Lecture Notes on Basic Celestial Mechanics: https://sirrah.troja.mff.cuni.cz/~davok/Sergei_Klioner_skripta.pdf

**University course notes**
- MIT OCW 8.033 Schwarzschild notes: https://ocw.mit.edu/courses/8-033-relativity-fall-2006/5dae4a0c2d69b367951ddea5e17a2cea_schwarzschild.pdf
- U. Maryland astr498 lectures 10 & 12: https://pages.astro.umd.edu/~mcmiller/teaching/astr498/lecture10.pdf · https://pages.astro.umd.edu/~mcmiller/teaching/astr498/lecture12.pdf
- U. Maryland ASTR630 Roche limit: https://pages.astro.umd.edu/~dphamil/ASTR630/handouts/RocheLimit.pdf
- Jacobson, Black Hole Thermodynamics (UMD): https://www.physics.umd.edu/grt/taj/776b/lectures.pdf
- Tübingen GTR notes (Kokkotas): https://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/GTR_files/GTR2018_3b.pdf
- Roma1 INFN GR notes ch. 21 (Kerr): https://www.roma1.infn.it/teongrav/onde19_20/kerr.pdf
- UNC Wilmington Kerr metric notes: https://people.uncw.edu/hermanr/BlackHoles/Kerr_Metric_II.pdf
- ASU MAE462 Lectures 4 & 7: https://control.asu.edu/Classes/MAE462/462Lecture04.pdf · https://control.asu.edu/Classes/MAE462/462Lecture07.pdf
- ANU High Energy Astrophysics Lecture 8 (Bicknell): https://www.mso.anu.edu.au/~geoff/HEA/8_Relativistic_Effects.pdf
- Harvard Math 118 N-body notes: https://legacy-www.math.harvard.edu/archive/118r_spring_05/handouts/nbody.pdf
- Princeton Dynamics §3.4: https://www.princeton.edu/~wbialek/intsci_web/dynamics3.4.pdf
- U. Alabama Schwarzschild notes: https://pages.physics.ua.edu/staff/fabi/phRel/SWSolution.pdf
- UCSD Physics 161 (Griest), CMU 33-331, SFU PHYS 390 tidal-force notes: https://courses.physics.ucsd.edu/2011/Winter/physics161/p161.14feb11.pdf · https://www.andrew.cmu.edu/course/33-331/pm11.pdf · https://www.sfu.ca/~boal/390lecs/390lec8.pdf
- CSUN — the three anomalies: http://www.csun.edu/~hcmth017/master/node14.html
- Ohio State — IAU 2015 nominal constants: https://www.astronomy.ohio-state.edu/pogge.1/Ast2292/Docs/NominalConstants.pdf

**Other technical references**
- René Schwarz Memoranda 1 & 2 (elements ↔ state vectors): https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf · https://downloads.rene-schwarz.com/download/M002-Cartesian_State_Vectors_to_Keplerian_Orbit_Elements.pdf
- orbital-mechanics.space (classical elements, perifocal frame, state-vector inverse, barycenter, SOI): https://orbital-mechanics.space/
- Vallado-style RV2COE implementation: https://github.com/dinkelk/astrodynamics/blob/master/astrodynamics/RV2COE.m
- poliastro SOI docs: https://poliastro-py.readthedocs.io/en/latest/api/safe/threebody/soi.html
- CU Boulder Intro Orbital Mechanics ch. 6: https://colorado.pressbooks.pub/introorbitalmechanics/chapter/chapter-6-keplers-prediction-problem/
- Oregon State space systems notes: https://kyleniemeyer.github.io/space-systems-notes/orbital-mechanics/two-body-problems.html
- Physics LibreTexts (many-body angular momentum; Skidmore GR effective potential; point-mass lensing; quantum effects near BHs): https://phys.libretexts.org/
- MathPages — orbital proper time: https://www.mathpages.com/rr/s6-05/6-05.htm
- Tipler Modern Physics companion — Mercury perihelion: https://www.macmillanlearning.com/studentresources/college/physics/tiplermodernphysics6e/more_sections/more_chapter_2_1-perihelion_of_mercurys_orbit.pdf
- nuclear-power.com — Bekenstein–Hawking entropy: https://www.nuclear-power.com/bekenstein-hawking-entropy/
- V. Toth Hawking calculator · iCalculator BH temperature (numeric cross-checks): https://www.vttoth.com/CMS/physics-notes/311-hawking-radiation-calculator · https://physics.icalculator.com/black-hole-temperature-calculator.html
- mathscinotes.com & astronomicalreturns.com (Roche limit, Hill spheres): https://www.mathscinotes.com/2016/03/roche-limit-examples/ · https://www.astronomicalreturns.com/2021/06/roche-limit-radius-of-disintegration.html · https://www.astronomicalreturns.com/2022/08/hill-spheres-where-moonmoons-are.html
- physicsfundamentals.org — escape velocity: https://physicsfundamentals.org/blog/escape-velocity
- Physics Forums (Hill sphere derivation; radial infall proper time): https://www.physicsforums.com/threads/complete-hill-sphere-derivation-learn-the-equation-and-its-derivation-process.197797/ · https://www.physicsforums.com/threads/free-fall-from-rest-at-a-particular-radius-to-the-central-singularity.1079546/
- Grokipedia (Roche limit; ISCO — secondary cross-checks): https://grokipedia.com/page/Roche_limit · https://grokipedia.com/page/Innermost_stable_circular_orbit
- EPFL Wikispeedia archived Hawking radiation article (evaporation coefficient): https://dlab.epfl.ch/wikispeedia/wpcd/wp/h/Hawking_radiation.htm

**Wikipedia (each cross-checked against at least one independent source above)**
- Center of mass · Escape velocity · Hill sphere · Sphere of influence (astrodynamics) · Tidal force · Eccentric anomaly · Perifocal coordinate system · Vis-viva equation · Mean motion · Tests of general relativity · Schwarzschild geodesics · Gravitational redshift · Shapiro time delay · Schwarzschild metric · Photon sphere · Innermost stable circular orbit · Kerr metric · Lense–Thirring precession · Hawking radiation · Schwarzschild radius · Sagittarius A* · Relativistic Doppler effect · Relativistic beaming · Accretion disk — all at https://en.wikipedia.org/wiki/