This calculator collects several advanced applications of gravity: black hole event horizons, photon spheres,
Roche limits, tidal forces, gravity assists, and a simple orbital decay estimate.
These topics use different formulas, but they all come from the same central idea: gravity changes motion, energy, and structure.
1. Schwarzschild radius
The Schwarzschild radius is the event horizon radius of a non-rotating, uncharged black hole:
\[
\boxed{
R_s=\frac{2GM}{c^2}
}.
\]
Here \(G\) is the gravitational constant, \(M\) is the mass, and \(c\) is the speed of light.
If an object of mass \(M\) is compressed inside \(R_s\), the escape speed at that radius reaches the speed of light.
2. Sun collapsed to a black hole
For the Sun,
\[
M_{\odot}\approx1.98847\times10^{30}\ \mathrm{kg}.
\]
Then
\[
R_s=
\frac{2(6.67430\times10^{-11})(1.98847\times10^{30})}
{(2.99792458\times10^8)^2}.
\]
\[
R_s\approx2.95\times10^3\ \mathrm{m}.
\]
Therefore,
\[
\boxed{
R_s\approx2.95\ \mathrm{km}
}.
\]
This is why the common estimate says that the Sun’s Schwarzschild radius is about \(3\ \mathrm{km}\).
3. Photon sphere and ISCO
In Schwarzschild geometry, the photon sphere is located at
\[
\boxed{
r_{\mathrm{photon}}=\frac{3}{2}R_s
}.
\]
It is the radius where photons can, ideally, orbit the black hole. This orbit is unstable.
The innermost stable circular orbit for massive particles around a non-rotating black hole is
\[
\boxed{
r_{\mathrm{ISCO}}=3R_s
}.
\]
Inside the ISCO, stable circular orbits are not possible in the Schwarzschild model.
4. Roche limit
The Roche limit estimates the distance at which tidal forces from a primary body can tear apart a secondary body.
A common density-based approximation is
\[
\boxed{
d_{\mathrm{Roche}}
\approx
kR
\left(
\frac{\rho_M}{\rho_m}
\right)^{1/3}
}.
\]
Here:
- \(R\) is the radius of the primary body,
- \(\rho_M\) is the density of the primary body,
- \(\rho_m\) is the density of the secondary body,
- \(k\) is a coefficient depending on the satellite model.
For a fluid satellite, a common value is
\[
k\approx2.44.
\]
For a more rigid satellite, a rough value is
\[
k\approx1.26.
\]
5. Tidal acceleration difference
Tidal effects happen because gravity is stronger on the near side of an object than on the far side.
For an object of length \(L\) at distance \(r\) from a mass \(M\), the approximate gravitational acceleration difference is
\[
\boxed{
\Delta g\approx\frac{2GML}{r^3}
}.
\]
This formula shows the strong distance dependence: tidal effects scale as \(1/r^3\).
Moving twice as far away reduces the tidal difference by about a factor of \(8\).
6. Gravitational slingshot
A gravitational slingshot, or gravity assist, changes a spacecraft’s heliocentric velocity by using the motion of a planet.
In the planet’s frame, the spacecraft enters and leaves with the same hyperbolic excess speed \(v_{\infty}\), but its direction changes.
In the star’s frame, the planet’s velocity is added vectorially:
\[
\mathbf v_{\mathrm{helio}}
=
\mathbf V_{\mathrm{planet}}
+
\mathbf v_{\infty}.
\]
Because the outgoing \(v_{\infty}\) direction differs from the incoming direction, the heliocentric speed can increase or decrease.
The calculator uses a simplified vector model:
\[
\Delta v_{\mathrm{helio}}
=
|\mathbf V+\mathbf v_{\infty,\mathrm{out}}|
-
|\mathbf V+\mathbf v_{\infty,\mathrm{in}}|.
\]
A positive value means the spacecraft gains heliocentric speed.
A negative value means it loses heliocentric speed.
7. Energy change in a slingshot
The specific kinetic energy change in the heliocentric frame is
\[
\boxed{
\Delta \varepsilon
=
\frac12
\left(
v_{\mathrm{out}}^2-v_{\mathrm{in}}^2
\right)
}.
\]
This is energy per unit spacecraft mass.
The planet loses or gains a tiny corresponding amount of orbital energy, but because the planet is so massive, its speed change is extremely small.
8. Simple orbital decay from drag
For a spacecraft in a low orbit, atmospheric drag removes mechanical orbital energy.
A simple drag model is
\[
F_D=\frac12\rho C_DA v^2.
\]
The power lost to drag is
\[
P_D=F_Dv.
\]
For a circular orbit,
\[
E=-\frac{GMm}{2r}.
\]
Therefore,
\[
\frac{dE}{dr}
=
\frac{GMm}{2r^2}.
\]
If drag removes energy at rate \(P_D\), then a rough radial decay rate is
\[
\boxed{
\frac{dr}{dt}
\approx
-\frac{P_D}{dE/dr}
}.
\]
This is only a rough preview because real atmospheric density changes rapidly with altitude and solar activity.
9. Event horizon, photon sphere, and ISCO summary
| Feature |
Formula |
Meaning |
| Event horizon |
\(R_s=2GM/c^2\) |
Boundary from which light cannot escape in the Schwarzschild model |
| Photon sphere |
\(r=1.5R_s\) |
Unstable circular photon orbit |
| ISCO |
\(r=3R_s\) |
Innermost stable circular orbit for massive particles around a non-rotating black hole |
10. Tidal and Roche summary
| Goal |
Formula |
Use |
| Fluid Roche limit |
\(d\approx2.44R(\rho_M/\rho_m)^{1/3}\) |
Disruption distance for a deformable body |
| Rigid Roche estimate |
\(d\approx1.26R(\rho_M/\rho_m)^{1/3}\) |
Lower disruption estimate for a stronger body |
| Tidal acceleration difference |
\(\Delta g\approx2GML/r^3\) |
Stretching acceleration across length \(L\) |
11. Gravity assist summary
| Quantity |
Expression |
Meaning |
| Incoming heliocentric velocity |
\(\mathbf V+\mathbf v_{\infty,\mathrm{in}}\) |
Spacecraft velocity before the flyby in the star frame |
| Outgoing heliocentric velocity |
\(\mathbf V+\mathbf v_{\infty,\mathrm{out}}\) |
Spacecraft velocity after the flyby in the star frame |
| Speed change |
\(\Delta v=|\mathbf v_{\mathrm{out}}|-|\mathbf v_{\mathrm{in}}|\) |
Approximate heliocentric speed gain or loss |
12. Assumptions and limits
- The black hole formulas assume a non-rotating, uncharged Schwarzschild black hole.
- Rotating black holes require Kerr geometry, which changes the ISCO and horizon structure.
- Roche limits depend on internal strength, spin, orbit eccentricity, and material behavior.
- The slingshot model is a simplified vector model and does not solve the full flyby trajectory.
- The orbital decay model assumes circular motion and simple drag, so it is only a rough preview.
- Real mission design requires numerical propagation, atmosphere models, and relativistic corrections where needed.
Key idea: advanced gravitation combines geometry, energy, and tidal gradients. The same gravitational field can create event horizons, tear objects apart, redirect spacecraft, or slowly drain orbital energy.
