--- title: "The Two-Body Problem, Orbital Energy and Kepler's First Two Laws" description: "Two bodies bound by gravity both orbit their common centre of mass, and the problem reduces to one body moving in a central field. Conservation of angular momentum gives the equal-areas law directly, " canonical: https://lightmysky.com/learn/science/the-two-body-problem-orbital-energy-and-keplers-first-two-laws-mt_CjHFftkB6M source: https://lightmysky.com/learn/science/the-two-body-problem-orbital-energy-and-keplers-first-two-laws-mt_CjHFftkB6M.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Two-Body Problem, Orbital Energy and Kepler's First Two Laws Two bodies bound by gravity both orbit their common centre of mass, and the problem reduces to one body moving in a central field. Conservation of angular momentum gives the equal-areas law directly, and the sign of the total energy decides whether the path closes into an ellipse or escapes. Subject: Science · Area: Forces & Motion · Ages 20 to 21 Page: https://lightmysky.com/learn/science/the-two-body-problem-orbital-energy-and-keplers-first-two-laws-mt_CjHFftkB6M ## Ready when they can - Derives the equal-areas law from angular momentum conservation rather than quoting it as an observation - Classifies an orbit as bound or unbound from the sign of the total energy and finds the speed at any radius - Explains why the Sun also moves, and how that wobble is what a radial-velocity planet search measures ## Lesson: Equal areas, bound orbits, wobbling suns Kepler second law says the line from Sun to planet sweeps equal areas in equal times. This is angular momentum conservation in disguise. Gravity pulls straight inward, so it cannot twist the motion, and the areal speed stays fixed. A planet therefore races near perihelion and lingers near aphelion. The same argument covers any central force, from comets to binary stars. Total orbital energy is kinetic plus gravitational potential, with potential set to zero at infinity. Its sign sorts every trajectory. Negative means bound and looping, zero means just barely escaping, and positive means unbound and leaving for good. This one sign test classifies circles, ellipses, parabolas, and hyperbolas. Energy conservation then gives the speed at any radius without solving the full path. **Example.** Two bodies both orbit their shared centre of mass, so the Sun moves too. Its small wobble is exactly what radial velocity planet searches measure. Watch starlight shift red and blue in a steady rhythm as unseen planets tug their star around. From the shift period and size, astronomers infer the hidden planet. **Tip.** Measure position and velocity once, compute the total energy, and you know whether the object stays or goes plus how fast it moves anywhere else. Bound orbits trade kinetic for potential and back as they loop. Raising an orbit costs energy, which is why transfer orbits are planned the way they are. **Recap.** Central pull conserves angular momentum into equal areas, the energy sign sorts bound from escaping, and the shared centre of mass makes stars wobble for planet hunters. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Angular Momentum and Its Conservation](https://lightmysky.com/learn/science/angular-momentum-and-its-conservation-mt_9v3S9-rhwc) - [Gravitational Potential and Escape Speed](https://lightmysky.com/learn/science/gravitational-potential-and-escape-speed-mt_drqTwCD9W8) - [The Centre of Mass and the Motion of a System](https://lightmysky.com/learn/science/the-centre-of-mass-and-the-motion-of-a-system-mt_fHO_KrMNKo) - [Orbits: Period, Speed and Kepler's Third Law](https://lightmysky.com/learn/science/orbits-period-speed-and-keplers-third-law-mt_PLKLykGF8Z) - [Finding Exoplanets](https://lightmysky.com/learn/science/finding-exoplanets-mt_UjuriPLVgT)