Gravity and Orbits

Useful constants: $G = 6.67 \times 10^{-8}~\mathrm{cm^3\,g^{-1}\,s^{-2}}$ ; $M_\odot = 2 \times 10^{33}~\mathrm{g}$ ; $R_\odot = 7 \times 10^{10}~\mathrm{cm}$ ; $M_\oplus = 6 \times 10^{27}~\mathrm{g}$ ; $R_\oplus = 6.4 \times 10^{8}~\mathrm{cm}$ ; $M_{\mathrm{Moon}} = 7.3 \times 10^{25}~\mathrm{g}$ ; $R_{\mathrm{Moon}} = 1.7 \times 10^{8}~\mathrm{cm}$ ; $1~\mathrm{AU} = 1.5 \times 10^{13}~\mathrm{cm}$ ; $1~\mathrm{yr} = 3.16 \times 10^{7}~\mathrm{s}$ .

Show explicit units, and for each result run a sanity check. Worked solutions are released after the homework due date.

Conceptual

Problem

⭐⭐ Empirical vs. physical laws. Kepler discovered $P^2 \propto a^3$ ; Newton explained why it must be true.

(a) What makes Kepler’s Third Law an empirical law rather than a physical law?
(b) If you observed $P^2 \propto a^4$ in another planetary system, what would that imply about the gravitational force law there?
(c) Newton’s law is universal. What predictive power does that universality add that Kepler’s empirical statement lacks?

Problem

⭐⭐ The centripetal-force misconception. A student claims: “A planet in orbit experiences two forces — gravity pulling it toward the Sun, and a centripetal force keeping it moving in a circle.”

(a) Identify exactly what is wrong with this statement.
(b) Rewrite it correctly, naming how many real forces act and what role each plays.
(c) If someone instead invokes a centrifugal force, under what circumstances (if any) is that language valid?

Problem

⭐⭐ Energy signs and orbital fate. The total mechanical energy is $E = \tfrac{1}{2}mv^2 - GMm/r$ .

(a) Why is the gravitational potential energy negative (with zero defined at infinity)?
(b) What do $E < 0$ , $E = 0$ , and $E > 0$ mean physically for the orbit type?
(c) A comet approaches the Sun with $E > 0$ . A student says, “It has positive energy, so it’s gaining energy as it approaches.” Correct this.

Calculation

Problem

⭐⭐ Orbital velocity practice. Using $v_{orb} = \sqrt{GM/r}$ :

(a) Calculate Earth’s orbital velocity around the Sun. Express in cm/s and km/s.
(b) Show explicitly that $\sqrt{GM/r}$ carries units of velocity.
(c) Mars orbits at $1.52$ AU. Using a ratio ( $v \propto r^{-1/2}$ ), find Mars’s orbital velocity.
(d) Sanity check: should Mars move faster or slower than Earth?

Problem

⭐⭐ Escape velocity comparison. Using $v_{esc} = \sqrt{2GM/R}$ :

(a) Calculate Earth’s escape velocity (in cm/s and km/s).
(b) The Moon has $M_{\mathrm{Moon}} = 7.3 \times 10^{25}$ g and $R_{\mathrm{Moon}} = 1.7 \times 10^8$ cm. Calculate its escape velocity.
(c) By what factor is Earth’s escape velocity larger than the Moon’s?
(d) Interpret: why did Apollo astronauts need a smaller rocket to leave the Moon?

Problem

⭐⭐ Kepler III with Newton. Newton’s version is $P^2 = \dfrac{4\pi^2}{G(M+m)}a^3$ .

(a) For Earth, $m \ll M_\odot$ . Write the simplified form.
(b) Verify it gives $P \approx 1$ year for $a = 1$ AU.
(c) Confirm both sides have units of $\mathrm{s^2}$ .
(d) A binary star has two equal $1\,M_\odot$ stars. How does its Kepler “constant” $P^2/a^3$ compare to the Sun–Earth system?

Problem

⭐⭐ Bound or unbound? A comet passes through Earth’s orbit ( $r = 1$ AU) moving at $50$ km/s. Decide whether the comet is bound (returns on a closed orbit), exactly parabolic (marginally bound), or unbound (escapes the Solar System).

(a) Compute the escape velocity from the Sun at $r = 1$ AU using $v_{esc} = \sqrt{2}\,v_{orb}$ , and compare it with $50$ km/s.
(b) Sanity check: how does $50$ km/s compare with Earth’s orbital velocity at the same radius?

Synthesis

Problem

⭐⭐ Exoplanet orbital velocities. Use Earth as reference ( $M = M_\odot$ , $r = 1$ AU) and $v \propto \sqrt{M/r}$ . Express answers in units of $v_\oplus$ .

(a) A hot Jupiter orbits at $r = 0.05$ AU around a Sun-like star ( $M = M_\odot$ ). Find $v/v_\oplus$ .
(b) A planet orbits a red dwarf ( $M = 0.25\,M_\odot$ ) at $r = 0.25$ AU. Find $v/v_\oplus$ .
(c) Sanity check: do your two answers make physical sense?

Problem

⭐⭐ (Challenge) Complete workflow: weighing Jupiter. Io orbits Jupiter with period $P = 1.77$ days and semi-major axis $a = 4.22 \times 10^{10}$ cm.

(a) Use Newton’s Kepler III ( $m \ll M$ ) to solve for Jupiter’s mass $M$ (in grams).
(b) Unit check: verify $\dfrac{a^3}{GP^2}$ has units of mass.
(c) Express your answer in grams and in solar masses.
(d) What observable quantities did you need?

Problem

⭐⭐ The $\sqrt{2}$ factor. Escape velocity is $v_{esc} = \sqrt{2}\,v_{orb}$ at any radius.

(a) Derive this from $v_{orb} = \sqrt{GM/r}$ and $v_{esc} = \sqrt{2GM/r}$ .
(b) A spacecraft in a circular orbit wants to escape. By what factor must it increase its speed?
(c) Explain the energy interpretation: why does escape require adding kinetic energy equal to the orbital kinetic energy?

Problem

⭐⭐ Virial theorem capstone. A star cluster has total kinetic energy $K = 1.0 \times 10^{50}$ erg and is in virial equilibrium ( $2K + U = 0$ ).

(a) Use the virial theorem to find its gravitational potential energy $U$ (in erg, with sign).
(b) Compute the total energy $E = K + U$ and state whether the cluster is bound.
(c) A tidal interaction injects an additional $0.6 \times 10^{50}$ erg of kinetic energy without changing the cluster size. What happens to the sign of $E$ ? Does the cluster remain bound?