Gravitation and orbits

So far gravity has been the constant downward force mg. That is an excellent approximation near the Earth's surface, but it hides one of the great unifications of physics: the force that makes an apple fall also holds the Moon, the planets and every satellite in orbit. Newton's law of universal gravitation explains why g is 9.81 m/s² at the surface and less at altitude, how fast a satellite must travel to stay in orbit, why astronauts float, how much energy it takes to escape the Earth, and how spacecraft move between orbits. Satellites now underpin navigation (GPS and its counterparts), communications, weather forecasting and Earth observation, so orbital mechanics has become everyday engineering. This lesson develops the law of gravitation, gravitational potential energy, circular and elliptical orbits, Kepler's laws and the transfer between orbits.

Newton's law of gravitation

Every two masses attract each other with a force proportional to the product of their masses and inversely proportional to the square of the distance between their centres:

The gravitational constant G is tiny, so gravity between everyday objects is negligible: two 1000 kg cars 3 m apart attract each other with about 7 µN. Only when one mass is enormous, like the Earth (5.97 × 10²⁴ kg), does gravity dominate. For a spherical body, the force outside it is the same as if all its mass were at its centre, a result Newton proved with the calculus he invented for the purpose (and that Gauss's law, MATH-209, makes easy).

At the Earth's surface, \(r = R_E = 6371\) km, and the force on a mass m is mg, so

The product \(GM_E = 3.986 \times 10^{14}\) m³/s² is known far more precisely than G or \(M_E\) separately, and is what orbit calculations use. (The measured g also varies slightly with latitude because of the Earth's rotation and shape, from about 9.78 m/s² at the equator to 9.83 at the poles, and with local geology, which geophysicists use to find ore bodies.)

Gravitational potential energy

The potential energy mgh applies only near the surface, where g is constant. In general, with zero potential energy taken at infinite separation, the gravitational potential energy of a mass m at distance r from the centre of a planet of mass M is

It is negative everywhere and rises toward zero as r increases: work must be done to separate the masses. Near the surface, the difference \(U(R_E + h) - U(R_E)\) reduces to mgh for small h, as Lesson 5 used. Setting the kinetic energy at launch equal to the energy needed to reach infinity gives the escape speed:

For the Earth this is 11.2 km/s, about 40 000 km/h, ignoring air resistance; for the Moon it is only 2.4 km/s, which is why lunar missions could return with a much smaller rocket than the one that left Earth.

Circular orbits

A satellite in a circular orbit of radius r moves at constant speed, and gravity provides exactly the centripetal force (Lesson 4): \(GMm/r^2 = mv^2/r\). The orbital speed and period follow:

Neither depends on the satellite's mass. The square of the period is proportional to the cube of the radius, which is Kepler's third law, found from planetary observations decades before Newton explained it. Higher orbits are slower and have longer periods.

The energy of a satellite in a circular orbit is the sum of its kinetic energy \(\tfrac{1}{2}mv^2 = GMm/(2r)\) and its potential energy \(-GMm/r\):

Raising a satellite to a higher orbit increases its total energy (makes it less negative) even though it then moves more slowly. Conversely, atmospheric drag on a low satellite removes energy, so the satellite drops to a lower orbit and, paradoxically, speeds up; the ISS needs periodic reboosts for this reason.

Elliptical orbits and Kepler's laws

Circular orbits are a special case. In general, a body bound by gravity moves on an ellipse with the central mass at one focus (Kepler's first law). The closest point is the periapsis (perigee for the Earth) and the farthest the apoapsis (apogee). The line from the central mass to the orbiting body sweeps out equal areas in equal times (Kepler's second law), which is conservation of angular momentum (Lesson 8): the body moves fastest at periapsis and slowest at apoapsis. And Kepler's third law holds with r replaced by the semi-major axis a, half the longest diameter of the ellipse: \(T = 2\pi\sqrt{a^3/GM}\). The speed at any point on an elliptical orbit follows from energy conservation:

This "vis-viva" equation reduces to the circular-orbit speed when r = a.

Changing orbits

A spacecraft changes orbit by firing its engine to change its velocity, measured as delta-v (Δv). The most economical way to move between two circular orbits in the same plane is the Hohmann transfer: a first burn raises the far side of the orbit to the target radius, putting the craft on a half-ellipse, and a second burn at the far point circularizes the orbit. Delta-v is the currency of space missions, because the propellant needed grows exponentially with it (the rocket equation): every extra kilometre per second costs a large increase in launch mass.

import math

GM = 3.986e14                      # Earth, m^3/s^2
r1, r2 = 6.771e6, 4.2164e7         # low orbit (400 km altitude) and geostationary radius

v1 = math.sqrt(GM / r1)
a = (r1 + r2) / 2                  # semi-major axis of the transfer ellipse
vp = math.sqrt(GM * (2 / r1 - 1 / a))
va = math.sqrt(GM * (2 / r2 - 1 / a))
v2 = math.sqrt(GM / r2)
print(f"Circular speeds: {v1 / 1000:.3f} km/s low, {v2 / 1000:.3f} km/s geostationary")
print(f"Burn 1: {(vp - v1) / 1000:.3f} km/s, burn 2: {(v2 - va) / 1000:.3f} km/s, total {(vp - v1 + v2 - va) / 1000:.3f} km/s")
print(f"Transfer time (half an ellipse): {math.pi * math.sqrt(a**3 / GM) / 3600:.2f} h")

# Check by stepping the transfer orbit with the velocity Verlet method
x, y, vx, vy = r1, 0.0, 0.0, vp
dt, t, rmax = 2.0, 0.0, r1
ax, ay = -GM * x / r1**3, -GM * y / r1**3
while t < 20 * 3600:
    vx += 0.5 * dt * ax; vy += 0.5 * dt * ay
    x += dt * vx; y += dt * vy
    r = math.hypot(x, y)
    ax, ay = -GM * x / r**3, -GM * y / r**3
    vx += 0.5 * dt * ax; vy += 0.5 * dt * ay
    t += dt
    if r > rmax:
        rmax, t_apo = r, t
print(f"Simulated: farthest point {rmax / 1000:.0f} km from the middle of the Earth, reached after {t_apo / 3600:.2f} h")

Circular speeds: 7.673 km/s low, 3.075 km/s geostationary
Burn 1: 2.399 km/s, burn 2: 1.457 km/s, total 3.857 km/s
Transfer time (half an ellipse): 5.29 h
Simulated: farthest point 42164 km from the middle of the Earth, reached after 5.29 h

The two burns total about 3.9 km/s, and the transfer takes 5.3 hours. The numerical simulation, using the velocity Verlet method (MATH-300), reaches exactly the geostationary radius at the predicted time, confirming both the formulas and the code. In practice, satellites launched from Canada or the northern United States also need to change their orbital plane to the equator, which adds considerable delta-v; that is why launch sites near the equator are favoured for geostationary missions.

Navigation satellites show how precisely these ideas are used. GPS satellites orbit at about 20 200 km altitude with a period of almost exactly half a sidereal day, so their ground tracks repeat daily. A receiver finds its position from the travel times of signals from four or more satellites, so timing errors of a microsecond would mean position errors of 300 m. The satellite clocks must even be corrected for relativity: they gain about 38 microseconds per day relative to clocks on the ground, mainly because gravity is weaker at their altitude, an effect Newtonian mechanics cannot explain but which engineers must still build in.

Tides and other effects

Because gravity weakens with distance, the Moon pulls harder on the near side of the Earth than on its centre, and harder on the centre than on the far side. These differences stretch the oceans into two bulges, one facing the Moon and one opposite, so most coasts see two high tides a day; the Sun adds a smaller effect, giving larger spring tides when Sun and Moon align and smaller neap tides between. Tidal ranges reach 16 m in the Bay of Fundy, where resonance of the bay amplifies them (Lesson 11), and tidal power projects there and elsewhere tap the energy. Tidal effects of the same kind gradually slow the Earth's rotation and push the Moon farther away, by about 3.8 cm per year.