Self-Assessment Quiz: Kepler's Laws and the Two-Body Problem

Twenty questions to check your grasp of Kepler's laws, their derivation from gravity, conic sections, the six orbital elements, orbital period, and the anomalies. Answer each before opening the key. Aim for 16 or more. Use $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$ and $R_\oplus = 6{,}371\ \text{km}$.

Question 1

Kepler's first law says a planet's orbit is an ellipse with the Sun at:

A) the center of the ellipse B) one focus of the ellipse C) the far end of the major axis D) the empty focus

Question 2

Kepler's second law (equal areas in equal times) is a direct expression of the conservation of:

A) energy B) linear momentum C) angular momentum D) mass

Question 3

Kepler's third law states that:

A) $T \propto a$ B) $T^2 \propto a^3$ C) $T^3 \propto a^2$ D) $T \propto a^2$

Question 4

In the derivation of §8.2, the fact that gravity is a central force (always along $\mathbf{r}$) leads immediately to:

A) the orbit being a circle B) the specific angular momentum $\mathbf{h}$ being constant C) the eccentricity being zero D) the period being independent of the primary

Question 5

The orbit equation $r = p/(1 + e\cos\nu)$ describes:

A) only circular orbits B) only elliptical orbits C) any conic section (circle, ellipse, parabola, or hyperbola) D) the ground track of a satellite

Question 6

An orbit with eccentricity $e = 1$ is a:

A) circle B) ellipse C) parabola D) hyperbola

Question 7

For an elliptical orbit, the perigee radius is:

A) $a(1 + e)$ B) $a(1 - e)$ C) $a/e$ D) $ae$

Question 8

Which of the six orbital elements changes continuously as a spacecraft coasts along an unperturbed orbit?

A) the semi-major axis $a$ B) the inclination $i$ C) the true anomaly $\nu$ D) the eccentricity $e$

Question 9

The inclination $i$ of an orbit specifies:

A) the size of the orbit B) the tilt of the orbital plane relative to the reference (equatorial) plane C) where periapsis is within the plane D) where the spacecraft is right now

Question 10

The orbital period depends on:

A) the semi-major axis $a$ only B) the eccentricity $e$ only C) both $a$ and $e$ D) the true anomaly $\nu$

Question 11

A geostationary orbit has a period of about:

A) 90 minutes B) 12 hours C) 23 h 56 m (one sidereal day) D) exactly 24 h 00 m

Question 12

The mean motion $n$ is equal to:

A) $2\pi/T = \sqrt{\mu/a^3}$ B) $\sqrt{\mu/r}$ C) $\sqrt{2\mu/r}$ D) $\mu/(2a)$

Question 13

Kepler's equation is:

A) $v^2 = \mu(2/r - 1/a)$ B) $M = E - e\sin E$ C) $r = p/(1 + e\cos\nu)$ D) $T = 2\pi\sqrt{a^3/\mu}$

Question 14

Kepler's equation cannot be solved for $E$ with elementary algebra because it is:

A) a quadratic B) transcendental ($E$ appears both alone and inside a sine) C) a differential equation D) dimensionally inconsistent

Question 15 (True/False, justify)

"Two orbits with the same semi-major axis $a$ but different eccentricities have different periods." True or false? Justify in one sentence.

Question 16 (True/False, justify)

"The mean anomaly $M$ and the true anomaly $\nu$ are equal at every point of an eccentric orbit." True or false? Say where, if anywhere, they are equal.

Question 17 (True/False, justify)

"Kepler's laws are fundamental postulates that Newton assumed in order to build his theory of gravity." True or false? Explain the actual logical relationship.

Question 18 (Short answer)

A circular orbit has radius $r = 6{,}771\ \text{km}$. Compute its period in minutes, showing the calculation. (This is the ISS-like low orbit.)

Question 19 (Short answer)

An orbit has $r_p = 7{,}000\ \text{km}$ and $r_a = 21{,}000\ \text{km}$. Find its semi-major axis and eccentricity, and classify the conic.

Question 20 (Short answer)

List, in order, the three steps you would take to find a spacecraft's true anomaly given the time elapsed since perigee. Name the equation used at each step.

Answer Key

Q Ans Note
1 B The primary sits at one focus, off-center — Kepler's key break from circular models.
2 C Equal areas in equal times ⇔ constant $\mathbf{h} = \mathbf{r}\times\mathbf{v}$.
3 B $T^2 = (4\pi^2/\mu)a^3$.
4 B A central force gives zero torque, so $\dot{\mathbf{h}} = 0$; the orbit is then planar with constant areal velocity.
5 C It is the general polar equation of a conic with the focus at the origin.
6 C $e = 1$ is the parabola, the marginal escape ($\varepsilon = 0$).
7 B $r_p = a(1-e)$; apogee is $r_a = a(1+e)$.
8 C Only $\nu$ moves; $a, e, i, \Omega, \omega$ are constant in the two-body problem.
9 B Inclination is the tilt of the orbital plane from the equator.
10 A $T = 2\pi\sqrt{a^3/\mu}$ — eccentricity drops out entirely.
11 C GEO matches the sidereal day, $23\ \text{h}\ 56\ \text{m}$, not the 24-h solar day.
12 A $n = 2\pi/T = \sqrt{\mu/a^3}$, the uniform angular rate carrying the mean anomaly.
13 B $M = E - e\sin E$ links uniform time ($M$) to geometry ($E$).
14 B It is transcendental; solve it numerically (e.g., Newton's method).
15 False Period depends on $a$ only; same $a$ ⇒ same period regardless of $e$.
16 False $M = \nu$ only at perigee ($0^\circ$) and apogee ($180^\circ$); in between $M < E < \nu$ (outbound).
17 False The reverse: Kepler's laws are theorems Newton derived from inverse-square gravity — consequences, not postulates.
18 — $T = 2\pi\sqrt{6{,}771^3/\mu} = 2\pi\sqrt{7.79\times10^{5}} = 2\pi(882.5) = 5{,}545\ \text{s} = 92.4\ \text{min}$.
19 — $a = (7{,}000+21{,}000)/2 = 14{,}000\ \text{km}$; $e = (21{,}000-7{,}000)/(21{,}000+7{,}000) = 14{,}000/28{,}000 = 0.5$; an ellipse ($0
20 — (1) $M = n(t-t_p)$ with $n = \sqrt{\mu/a^3}$; (2) solve $M = E - e\sin E$ for $E$ (iterate); (3) $\tan(\nu/2) = \sqrt{(1+e)/(1-e)}\tan(E/2)$.

Topics to review by question

Questions Topic Section
1, 2, 3, 17 Kepler's three laws and their status as theorems §8.1–8.2
4 Derivation from gravity (angular momentum) §8.2
5, 6, 7, 19 The orbit as a conic section; apsides §8.3
8, 9 The six orbital elements §8.4
10, 11, 12, 18 Period and mean motion §8.5
13, 14, 15, 16, 20 Anomalies and Kepler's equation §8.6