Motion of orbiting satellites

A satellite held in orbit is in free fall: gravity is the only force, and it supplies exactly the centripetal pull that curves the path around the central body. Setting gravity equal to that requirement is the single idea behind every formula in this topic.

From it follow the circular orbital speed $v_{orb}=\sqrt{GM/r}$, the escape speed $v_{esc}=\sqrt{2GM/r}$, the total energy of a bound orbit $E=-\dfrac{GMm}{2r}$, and Kepler's third law $T^2\propto r^3$. None of these contain the satellite's own mass.

Five habits carry the topic: escape speed is $\sqrt{2}$ times orbital, not twice; the orbiting mass cancels; a bound orbit's energy is negative; gravity falls off as $GM/r^2$ rather than staying constant; and the period grows as $r^{3/2}$.

For a circular orbit, gravity is the centripetal force: $\dfrac{GMm}{r^2}=\dfrac{mv^2}{r}$. The satellite mass $m$ appears on both sides and cancels, leaving

• $v_{orb}=\sqrt{\dfrac{GM}{r}}$.

Because $m$ cancels, two satellites at the same radius move at the same speed no matter how heavy each one is. The larger gravitational pull on a heavier satellite is matched exactly by its larger inertia. The speed is set by the central mass $M$ and the radius $r$: a bigger $M$ means faster, a bigger $r$ means slower.

Circular orbital speed is $v_{orb}=\sqrt{GM/r}$. The satellite's own mass cancels; only the central mass and the radius matter.

To escape, a body needs just enough kinetic energy to climb out of the gravitational well to infinity, where both kinetic and potential energy reach zero. Setting $\tfrac{1}{2}mv^2-\dfrac{GMm}{r}=0$ gives

• $v_{esc}=\sqrt{\dfrac{2GM}{r}}=\sqrt{2}\,v_{orb}$.

So escape speed is only $\sqrt{2}\approx1.41$ times the circular orbital speed, not twice it. The number $2$ does appear, but in the energy: escaping needs twice the kinetic energy of the circular orbit, and since kinetic energy goes as $v^2$, doubling it raises the speed by only $\sqrt{2}$. A body already in a bound orbit is, by definition, moving slower than escape speed.

Escape speed is $\sqrt{2}$ times the circular speed at that radius. The factor $2$ lives in the energy; the speed rises by $\sqrt{2}$.

For a circular orbit the kinetic energy is $K=\tfrac{1}{2}mv^2=\dfrac{GMm}{2r}$, and the potential energy, with its zero taken at infinity, is $U=-\dfrac{GMm}{r}=-2K$. Adding them,

• $E=K+U=\dfrac{GMm}{2r}-\dfrac{GMm}{r}=-\dfrac{GMm}{2r}$.

The total energy is negative: the negative potential energy is twice the size of the kinetic energy, so the sum falls below zero. That negative sign is what marks the orbit as bound, energy must be added to reach infinity. A larger orbit has a less negative energy, so raising an orbit costs energy. A path that just barely escapes is parabolic, with $E=0$. Here $r$ is the circular-orbit radius; a general elliptical bound orbit replaces it with the semi-major axis, $E=-\dfrac{GMm}{2a}$.

A circular orbit has $E=-\dfrac{GMm}{2r}$, negative and equal in size to its kinetic energy. A larger orbit is less negative; escape is $E=0$.

Gravity is not a fixed background value. The field strength is $g(r)=\dfrac{GM}{r^2}$, an inverse-square law measured from the center. Doubling the distance quarters the field; at the altitude of a low orbit, only a little farther than the surface, gravity is still about ninety percent of its surface value. Astronauts feel weightless because they are in free fall, not because gravity has switched off, and the field never reaches zero at any finite distance.

The period follows from the speed: $T=\dfrac{2\pi r}{v_{orb}}=2\pi\sqrt{\dfrac{r^3}{GM}}$, which is Kepler's third law,

• $T^2=\dfrac{4\pi^2}{GM}\,r^3$, so $T^2\propto r^3$ and $T\propto r^{3/2}$.

Quadrupling the radius multiplies the period by $4^{3/2}=8$, not by $4$. Like the speed, the period contains the central mass but not the satellite's mass.

Gravity falls as $g=GM/r^2$, never constant and never zero. The period grows as $T\propto r^{3/2}$: a wider orbit is much slower.

Orbital problems hinge on a few clean distinctions: escape versus orbital speed, whether the satellite's mass matters, the sign of a bound orbit's energy, how gravity thins with distance, and the way the period scales as $r^{3/2}$.

• Escape vs orbitalTreating escape speed as twice the orbital speed, or assuming an orbiting body already moves fast enough to escape. Fix: $v_{esc}=\sqrt{2}\,v_{orb}$.
• Mass mattersThinking a heavier satellite orbits faster or slower. Fix: the orbiting mass cancels, so $v=\sqrt{GM/r}$ and $T$ depend only on the central mass and radius.
• Wrong signTaking a bound orbit's total energy as positive or zero, or reading a more negative energy as a higher orbit. Fix: $E=-\dfrac{GMm}{2r}$, negative; a larger orbit is less negative.
• Constant gravityTreating gravity as fixed, falling off linearly, or vanishing in orbit. Fix: $g=GM/r^2$, an inverse-square field that weakens but never reaches zero.
• Kepler powerScaling the period in proportion to the radius or with the wrong power. Fix: $T^2\propto r^3$, so $T\propto r^{3/2}$.

Escape is $\sqrt{2}$, the mass cancels, bound energy is negative, gravity is inverse-square, and the period grows as $r^{3/2}$. The whole topic comes down to those five habits.