3.5 Motions of Satellites and Spacecraft

Orbits and Spacecraft Trajectories

Getting a satellite into orbit or sending a probe to another planet requires a solid understanding of orbital mechanics. These concepts connect directly to the gravity and Kepler's laws you've already studied, but now applied to human-made objects instead of natural ones.

Satellite Orbit Placement

To stay in orbit, a satellite needs to be above most of Earth's atmosphere and moving at just the right speed. Too slow and it falls back to Earth; too fast and it flies away.

A rocket accelerates the satellite to orbital velocity, which depends on how high the orbit is:

$v = \sqrt{\frac{GM}{r}}$

where $G$ is the gravitational constant, $M$ is Earth's mass, and $r$ is the distance from Earth's center to the satellite.

Notice that $r$ is in the denominator. That means higher orbits actually require lower velocities. A satellite in low Earth orbit (LEO) travels at about 7.8 km/s, while one in geostationary orbit (about 35,786 km up) only needs around 3.1 km/s.

Once the rocket reaches the target altitude and speed, the satellite is released. Its velocity is perpendicular to the pull of gravity, so it continuously "falls around" Earth rather than falling straight down.

• A circular orbit keeps the satellite at roughly the same altitude throughout.
• An elliptical orbit varies in altitude. The farthest point from Earth is called apogee, and the closest point is perigee.

Escaping Earth's Gravity

To leave Earth entirely, a spacecraft must reach escape velocity, the minimum speed needed to break free of Earth's gravitational pull without further propulsion.

• At Earth's surface, escape velocity is about 11.2 km/s.
• Escape velocity decreases with distance from Earth. At the Moon's orbit, it drops to roughly 1.4 km/s.

Reaching these speeds takes enormous energy. Chemical rockets (like the Saturn V or Falcon 9) are the most common propulsion method. Most use staged rockets, which drop empty fuel tanks as they go, reducing the spacecraft's mass so the remaining fuel is more effective.

Launch sites are chosen strategically. Launching eastward near the equator takes advantage of Earth's rotational speed (about 0.46 km/s at the equator), giving the rocket a free velocity boost. That's why major launch sites like Cape Canaveral and the Guiana Space Centre sit at low latitudes.

Low Earth Orbit vs. Interplanetary Trajectories

These two types of missions have very different energy requirements and orbital shapes.

Low Earth Orbit (LEO):

• Altitude of 160–2,000 km above Earth's surface
• Examples: the International Space Station (~408 km) and the Hubble Space Telescope (~547 km)
• Orbits are mostly circular or slightly elliptical
• Requires less energy than interplanetary missions
• Still encounters trace atmospheric drag, so LEO satellites need periodic reboosts (the ISS gets reboosted roughly every 2–3 months)

Interplanetary Spacecraft:

• Must first reach escape velocity to leave Earth's gravitational influence
• Requires significantly more energy than LEO missions
• Trajectories are typically elliptical (for orbiting a target) or hyperbolic (for flybys, where the spacecraft doesn't stay)
• Examples: Voyager 1 and 2, New Horizons (hyperbolic flyby of Pluto), Cassini (orbited Saturn), Galileo (orbited Jupiter)
• Many interplanetary missions use gravitational assists (see below) to reach their destinations without carrying impossibly large amounts of fuel

Orbital Mechanics and Spacecraft Maneuvers

Several key concepts govern how spacecraft change orbits and navigate between planets.

Kepler's Laws apply to spacecraft just as they do to planets. A satellite in an elliptical orbit moves faster at perigee and slower at apogee, exactly as Kepler's second law (equal areas in equal times) predicts.

Delta-v ($\Delta v$) is the total change in velocity a spacecraft needs for a maneuver. Mission planners calculate the $\Delta v$ budget for an entire mission to determine how much fuel to carry. Every orbit change, course correction, and braking burn costs $\Delta v$.

Hohmann Transfer Orbit: The most fuel-efficient way to move between two circular orbits. It works in two steps:

1. Fire engines to enter an elliptical transfer orbit that just touches both the starting orbit and the target orbit.
2. When the spacecraft reaches the target orbit, fire engines again to circularize into the new orbit.

This method is slow compared to more direct paths, but it uses the least $\Delta v$.

Gravitational Slingshot (Gravity Assist): A spacecraft flies close to a planet and uses that planet's gravity and orbital motion to change its own speed and direction. The spacecraft "borrows" a tiny amount of the planet's orbital energy. Cassini, for example, used gravity assists from Venus (twice) and Earth before heading to Saturn, saving enormous amounts of fuel. Without gravity assists, many outer solar system missions would be impractical with current rocket technology.