AstronomyCosmic Scales: How Orbits Let Us Weigh the Stars
Lumi floats beside a glowing orrery, one hand sweeping a planet along its tilted elliptical track while a tiny scale beam balances a distant star.
- Describe how Newton's law of universal gravitation sets the force between two masses.
- State Kepler's three laws and connect each to the shape, speed, and period of an orbit.
- Calculate a central body's mass from an orbiting object's period and orbital distance.
- Explain why a more distant planet takes longer to complete one orbit.
Key terms
- Universal gravitation
- Newton's law that every pair of masses attracts with a force proportional to their masses and inversely proportional to the square of their separation.
- Inverse-square law
- The rule that gravitational force falls off as one over the distance squared, so doubling separation cuts force to one-quarter.
- Kepler's third law
- The relation that an orbit's period squared is proportional to its semi-major axis cubed, encoding the central body's mass.
- Semi-major axis
- Half of an ellipse's longest diameter, used as the average orbital distance in Kepler's third law.
Newton's Synthesis of Kepler's Laws
Kepler's three laws were empirical patterns drawn from Tycho Brahe's data, but Newton showed they all follow from a single inverse-square gravitational force. The force F = Gm₁m₂/r² produces closed elliptical orbits (first law), causes a planet to sweep equal areas in equal times because angular momentum is conserved (second law), and, when combined with circular-orbit dynamics, yields T² ∝ a³ (third law). This unification turned descriptive astronomy into predictive physics: the same equation that drops an apple also governs the Moon.
Weighing the Cosmos from Motion
The full form of Kepler's third law, T² = 4π²a³/(G·M), shows that the central mass M is the only unknown once an orbiting body's period T and semi-major axis a are measured. By timing how long a planet, moon, or companion star takes to circle and how far it orbits, astronomers solve for M directly. This is how the Sun's mass, the masses of exoplanet host stars, and even the supermassive black hole at our galaxy's center have all been weighed, entirely from observed motion rather than any scale.
Worked examples
Two planets orbit one star; planet B is four times as far as planet A. How much longer is B's period?
- Apply Kepler's third law: T² ∝ a³, so T ∝ a^(3/2).
- The distance ratio is 4, so the period ratio is 4^(3/2).
- Evaluate: 4^(3/2) = (4^(1/2))³ = 2³ = 8.
Answer: Planet B's orbital period is about 8 times longer than planet A's.
Activity
Order these four planets from shortest to longest orbital period using Kepler's third law.
Practice
A satellite at distance r feels gravitational force F. Calculate the force when it moves to distance 2r and explain the factor using the inverse-square law.
Explain why astronomers can determine a star's mass from a planet's orbit but cannot directly read mass from the star's apparent brightness.
Common mistakes to avoid
- Doubling distance halves gravitational force.Gravity follows the inverse-square law, so doubling distance reduces the force to one-quarter, not one-half.
- A brighter star must be more massive.Apparent brightness depends on distance and luminosity, which relate to mass only through stellar models, not by direct measurement.
Check your understanding
Newton's law of gravitation says that if you double the distance between two masses, the gravitational force between them becomes:
A planet moves fastest in its orbit when it is:
Why can astronomers determine the Sun's mass from Earth's orbit but NOT directly from how bright the Sun looks?
Two planets orbit the same star. Planet B is four times as far from the star as Planet A. Using T^2 proportional to a^3, Planet B's orbital period is about:
Recap
Newton's inverse-square law F = Gm₁m₂/r² unifies Kepler's three laws, and because Kepler's third law T² ∝ a³ embeds the central mass, astronomers can weigh stars, planets, and black holes purely from an orbiting body's period and distance.
Reflect
How does it change your view of astronomy to realize we can weigh objects light-years away using nothing but their motion?