# Gravitation Exam Models - Orbit Transfer, Geostationary, Satellite Chase, Binary Stars

> Gravitation exam models: orbit comparison, GM = gR², geostationary orbit, cosmic speeds, orbit transfer, satellite chase, binary stars, mass and density.

- URL: https://qilingtools.com/en/physics/celestial-exam
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

This tool gathers the standard gravitation exam models in one place. It compares v, ω, T and a for satellites at different radii (higher orbits are slower with longer periods), applies GM = gR², finds the geostationary height and the cosmic speeds, and compares an object on the equator with near-surface and geostationary satellites. Animations show a low orbit → transfer ellipse → high orbit change, two satellites chasing each other, and binary and triple star systems. It also estimates a central body’s mass and density from an orbit or from surface gravity, always with the numbers substituted.

## How to use

1. Pick a tab: orbit comparison, equator vs LEO vs GEO, orbit transfer, satellite chase, binary / triple stars, or mass and density
2. Choose the central body or a classic case, then adjust altitudes, phase angle, masses or separation
3. Watch the orbit animation and charts, and read the results and step-by-step reasoning
4. Open Practice for random gravitation problems with instant marking

## Features

- Table of v, ω, T and a for several orbits plus a log chart against r: higher means slower and longer
- GM = gR², geostationary radius and height, first, second and third cosmic speeds
- Equator object vs near-surface vs geostationary satellite: v, ω, a and T with their rankings
- Animated Hohmann transfer: speed ranking at P and Q, equal accelerations at the same point, periods and energies
- Satellite chase with closest and farthest times and a distance–time graph; animated binary, linear and triangular triple stars
- Mass and density from an orbit or from surface gravity (near-surface ρ = 3π/(GT²)), plus random practice

## FAQ

### Why are higher orbits slower with longer periods?

For a satellite in a circular orbit gravity supplies the centripetal force: GMm/r² = mv²/r = mω²r = m(2π/T)²r = ma. So v = √(GM/r), ω = √(GM/r³), T = 2π√(r³/GM) and a = GM/r². The larger r is, the smaller v, ω and a are, and only T grows. This applies only to bodies held in orbit by gravity alone, not to an object resting on the rotating Earth.

### How do the speeds at P and Q compare during an orbit transfer?

Going from low circle 1 to a transfer ellipse to high circle 2: a burn at perigee P raises the speed to v_P above the circular speed v₁, gravity can no longer hold it on the circle and it moves out onto the ellipse. It slows to v_Q at apogee Q, where a second burn raises it to the circular speed v₂. So v_P > v₁ > v₂ > v_Q. At the same point, however, the acceleration a = GM/r² is the same on the ellipse and on the circle, and the higher orbit has more mechanical energy.

### How do an object on the equator, a near-surface satellite and a geostationary satellite compare?

The equatorial object and the geostationary satellite share the same period, so the same ω; with v = ωr and a = ω²r the satellite, being farther out, has larger v and a. The near-surface and geostationary satellites are both held by gravity alone, so the lower one is faster with larger ω and a and a shorter period. Altogether v_LEO > v_GEO > v_eq, a_LEO > a_GEO > a_eq and ω_LEO > ω_GEO = ω_eq. The equatorial object also feels the ground’s normal force, so GMm/R² = mv²/R does not apply to it.

### How do you find the radii and total mass of a binary star?

Their mutual gravity provides each star’s centripetal force and they circle the same point on the line joining them, with the same ω and period. Gm₁m₂/L² = m₁ω²r₁ = m₂ω²r₂, so m₁r₁ = m₂r₂: the radii (and speeds) are inversely proportional to the masses, r₁ = m₂L/(m₁ + m₂). Adding the equations gives G(m₁ + m₂) = ω²L³, so measuring T and L gives the total mass M = 4π²L³/(GT²), but not the individual masses.

### How can you estimate a planet’s mass and density?

From an orbit: measure a satellite’s orbital radius r and period T; GMm/r² = m(2π/T)²r gives M = 4π²r³/(GT²), and the satellite’s own mass cancels. Dividing by the volume gives ρ = 3πr³/(GT²R³), and for a near-surface orbit (r ≈ R) simply ρ = 3π/(GT²). From surface gravity: mg = GMm/R² gives M = gR²/G and ρ = 3g/(4πGR). The tool includes presets such as the Moon around the Earth, the Earth around the Sun and a near-surface satellite.

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