This unit explains why the Moon, planets, and satellites move the way they do, starting with Kepler's patterns and ending with Newton's and Einstein's explanations for them. By the end, you should be able to calculate gravitational force, field strength, orbital speed, and orbital period, and explain what makes an orbit fast or slow.
Read the big ideas first, then work through each topic's notes and try the check-yourself question before moving on.
Gravity is the same force whether it pulls an apple off a tree or holds Jupiter in orbit. This unit connects those two scales. You'll use the centripetal force ideas from circular motion and the energy and momentum ideas from earlier units, now applied to planets and satellites.
Honors Physics Unit 7 covers gravitation in two parts. First come Kepler's laws: orbits are ellipses with the Sun at one focus, planets sweep out equal areas in equal times, and is proportional to . Second is Newton's law of universal gravitation, , which you use to find gravitational force, field strength , circular orbital speed , and orbital period. The unit closes with how Einstein's general relativity describes gravity as the curvature of spacetime and where it improves on Newton.
Doubling the center-to-center distance cuts the force to one quarter, and tripling it cuts the force to one ninth. For spherical bodies, measure between their centers. An object on Earth's surface sits at m from Earth's center, not zero.
For a circular orbit around a much more massive body, gravity provides the centripetal force. Setting gives , so the satellite's mass cancels. Around the same central body, closer circular orbits are faster and have shorter periods.
General relativity treats gravity as mass and energy curving spacetime. In weak fields and at low speeds, its predictions are close to Newton's. It becomes necessary near black holes, for GPS clock corrections, and for small effects like the precession of Mercury's orbit.
Before Newton, falling objects and planetary orbits seemed to follow different rules. Newton's 1687 law of universal gravitation showed that the same inverse square attraction causes both, and Kepler's laws fall out of it as consequences. Einstein's theory changed the description of gravity, but it agrees closely with Newton's predictions when gravity is weak and speeds are low.
Kepler's three laws describe how planets move, and Newton later showed that all three follow from gravity.
Newton's law of universal gravitation says masses attract one another.
Kepler's three laws describe how planets move, and Newton later showed that all three follow from gravity. Johannes Kepler worked them out in the early 1600s using Tycho Brahe's careful observations of Mars.
The first law says orbits are ellipses with the Sun at one focus. For a noncircular ellipse, the other focus is empty; for a circle, both foci coincide at the center. Eccentricity measures how stretched the ellipse is. A circle has , and Earth's orbit () is very close to one.
The second law says a line from the Sun to a planet sweeps out equal areas in equal times. So a planet moves fastest at perihelion (its closest point) and slowest at aphelion (its farthest point). The reason is conservation of angular momentum. Gravity points straight at the Sun, so it exerts no torque about the Sun.
The third law links orbit size to orbit time with , where is the semi-major axis. Newton's version is , where is the mass of the central body. The orbiting body's mass drops out as long as it's much smaller than .
Worked example. Mars has a semi-major axis of AU. For an object much less massive than the Sun, measuring in years and in AU gives approximately . So , and years. The measured value is 687 days, or 1.88 years, so the law holds up.
Check yourself: A planet orbits a star with the same mass as the Sun at a semi-major axis of 4 AU. What is its orbital period? (Answer: , so years.)
Newton's law of universal gravitation says masses attract one another. For point masses or nonoverlapping spherical bodies with spherically symmetric mass distributions, , where N·m²/kg² and is the distance between their centers. If you double the distance, the force drops to one quarter, which is why it's called an inverse square law. The force is always attractive and equal on both objects, so Earth pulls on you exactly as hard as you pull on Earth.
Divide the force on a test mass by that mass to get gravitational field strength, measured in N/kg, equivalent to m/s². Outside a spherically symmetric body of mass , its field has magnitude . For a circular orbit around a much more massive body, gravity supplies the centripetal force, so gives .
Worked example. The ISS orbits about 420 km up, so m. Earth's m³/s². Then m/s², about 88% of surface gravity. Astronauts float because they and the station are falling together, not because gravity is gone. Their orbital speed is km/s.
Einstein's general relativity (1915) describes gravity as mass and energy curving spacetime instead of a force acting at a distance. It agrees closely with Newton in weak fields at low speeds. It also explains the extra precession of Mercury's orbit beyond Newtonian predictions and predicts how much light bends around massive objects.
Check yourself: What is the gravitational field strength at a point two Earth radii from Earth's center? (Answer: The distance doubled, so is one quarter of 9.8 m/s², about 2.45 m/s².)
| Question | Newton's gravity | Einstein's general relativity |
|---|---|---|
| What is gravity? | An attractive force between masses | The curvature of spacetime caused by mass and energy |
| How fast do gravitational effects travel? | Instantly (action at a distance) | At the speed of light, as gravitational waves |
| Where does it work well? | Weak fields and low speeds, like everyday motion and most orbits | In tested cases including strong gravity and high speeds, though it is not a complete theory of quantum gravity |
| Mercury's orbit | Predicts slightly too little precession | Matches the observed precession |
| Light passing a star | Predicts little or no bending | Correctly predicts gravitational lensing |
| Term | Definition |
|---|---|
| Newton's law of universal gravitation | The rule that masses attract one another. For point masses or nonoverlapping spherically symmetric bodies, the force has magnitude $F = G\frac{m_1 m_2}{r^2}$ and acts along the line between their centers. |
| Gravitational field strength | The gravitational force per unit mass at a point, measured in N/kg. Outside a spherically symmetric body, its field has magnitude $g = GM/r^2$. Near Earth's surface, $g$ is about 9.8 N/kg. |
| Kepler's first law | Each planet moves in an elliptical orbit with the Sun at one focus. |
| Kepler's second law | A line from the Sun to a planet sweeps out equal areas in equal times, so planets move faster when closer to the Sun. |
| Kepler's third law | The square of the orbital period is proportional to the cube of the semi-major axis, $T^2 \propto a^3$. |
| Orbital speed | The speed needed for a circular orbit, $v = \sqrt{GM/r}$, which gets smaller as the orbit gets larger. |
| Escape velocity | The minimum speed needed to reach arbitrarily far from an isolated body without further thrust. From Earth's surface, it is about 11.2 km/s when air resistance and other bodies' gravity are ignored. |
| General relativity | Einstein's 1915 theory that describes gravity as the curvature of spacetime by mass and energy. |
Problems often give altitude, like a satellite 420 km above Earth. Newton's law needs the center-to-center distance, so add Earth's radius first: m. Using the altitude alone gives a force that is hundreds of times too big.
Gravity depends on , not . Doubling the distance divides the force by 4, and tripling it divides by 9. Set up a ratio like to check yourself quickly.
At the ISS, gravity is still about 8.6 m/s², roughly 88% of what you feel on the ground. Astronauts feel weightless because they and the station are in free fall together, so nothing pushes up on them. Gravity is what keeps the station in orbit at all.
In and , is the central body's mass. These formulas assume the satellite is much less massive than the central body. A bolt and a space station in circular orbits at the same radius around the same body have the same orbital speed.