Kepler's laws describe how spacecraft, planets, and satellites move in orbit under gravity. For astronautics, these laws help engineers predict where a spacecraft will be, how fast it will move, and how long one orbit will take. The central idea is that an orbit is usually an ellipse, not a perfect circle, with the attracting body at one focus.
This is why a spacecraft speeds up near Earth and slows down when it is farther away.
The three laws connect the shape of an orbit, the changing speed along the path, and the relationship between orbital size and orbital period. Kepler's second law says a line from Earth to the spacecraft sweeps out equal areas in equal times, which is a direct sign of angular momentum conservation. Kepler's third law says larger orbits take longer, following T^2 = (4π^2/GM)a^3 for an object orbiting a much more massive body.
These ideas are used to plan satellite missions, transfer orbits, planetary flybys, and communication coverage.
Key Facts
- Kepler's First Law: An orbit is an ellipse with the central body at one focus.
- Kepler's Second Law: Equal areas are swept out in equal times, so the spacecraft moves faster near periapsis and slower near apoapsis.
- Kepler's Third Law: T^2 = (4π^2/GM)a^3 for a small object orbiting a much larger mass M.
- For circular orbits, v = sqrt(GM/r) and T = 2πsqrt(r^3/GM).
- Periapsis is the closest point in an orbit, and apoapsis is the farthest point.
- The semi-major axis a controls the orbital period more strongly than the ellipse's exact shape.
Vocabulary
- Ellipse
- An oval-shaped path where the sum of the distances from any point on the path to two fixed points called foci is constant.
- Focus
- One of two fixed points that define an ellipse, with the central attracting body located at one focus for a Keplerian orbit.
- Semi-major axis
- Half the longest width of an ellipse, represented by a, and the key distance used in Kepler's third law.
- Orbital period
- The time T required for a spacecraft or satellite to complete one full orbit.
- Periapsis
- The point in an orbit where the spacecraft is closest to the central body and moving fastest.
Common Mistakes to Avoid
- Putting Earth at the center of every elliptical orbit, which is wrong because Kepler's first law places the central body at one focus, not usually at the ellipse's center.
- Assuming orbital speed is constant in an ellipse, which is wrong because the spacecraft moves faster near periapsis and slower near apoapsis.
- Using the diameter instead of the semi-major axis in T^2 = (4π^2/GM)a^3, which gives an incorrect period because a is half the longest width of the ellipse.
- Thinking a larger orbit has only a slightly larger period, which is wrong because the period scales as a^(3/2), so increasing orbital size can greatly increase orbital time.
Practice Questions
- 1 A satellite is in a nearly circular orbit of radius 7.0 x 10^6 m around Earth. Using GM = 3.986 x 10^14 m^3/s^2, calculate its orbital speed with v = sqrt(GM/r).
- 2 A spacecraft has an elliptical Earth orbit with semi-major axis a = 1.20 x 10^7 m. Using T = 2πsqrt(a^3/GM) and GM = 3.986 x 10^14 m^3/s^2, find its orbital period in seconds and minutes.
- 3 A spacecraft travels from apoapsis toward periapsis in an elliptical orbit. Explain how its speed changes and connect your explanation to Kepler's second law.