Self-Assessment Quiz: Perturbations

Twenty questions on why real orbits drift, the J2 secular effects, drag, solar radiation pressure, third-body perturbations, and station-keeping. Answer each before opening the key. Aim for 16 or more. Use $J_2 = 1.0826\times10^{-3}$ and $g_0 = 9.81\ \text{m/s}^2$ where needed.

Question 1

A perturbation whose effect on an orbital element grows steadily with time, without averaging away, is:

A) periodic B) secular C) osculating D) resonant

Question 2

$J_2$ is the leading measure of Earth's:

A) magnetic field B) atmospheric density C) equatorial bulge (oblateness) D) rotation rate

Question 3

For a prograde orbit ($i < 90^\circ$), J2 nodal regression makes the ascending node drift:

A) eastward B) westward C) not at all D) toward the equator

Question 4

The J2 nodal-regression rate of an orbit at inclination $i = 90^\circ$ (polar) is:

A) maximum B) zero C) eastward D) equal to the apsidal rate

Question 5

Sun-synchronous orbits are typically found near which inclination?

A) $0^\circ$ B) $51.6^\circ$ C) $63.4^\circ$ D) $98^\circ$

Question 6

The critical inclination, at which J2 apsidal precession vanishes, is:

A) $28.5^\circ$ B) $63.4^\circ$ C) $90^\circ$ D) $98.6^\circ$

Question 7

To secular (accumulating) order, J2 changes which orbital elements?

A) $a$ and $e$ B) $\Omega$ and $\omega$ C) $i$ and $\nu$ D) all six

Question 8

The steady shrinking of a low orbit, ending in re-entry, is caused chiefly by:

A) solar radiation pressure B) third-body gravity C) atmospheric drag D) J2 oblateness

Question 9

As drag acts on an eccentric orbit, the orbit first becomes:

A) more eccentric B) more circular C) more inclined D) higher

Question 10

Two satellites at the same altitude have ballistic coefficients $\beta_1 = 50$ and $\beta_2 = 150\ \text{kg/m}^2$. Which decays faster?

A) satellite 1 (lower $\beta$) B) satellite 2 (higher $\beta$) C) they decay equally D) neither decays

Question 11

The radiation pressure of fully absorbed sunlight at $1\ \text{AU}$ ($S = 1{,}361\ \text{W/m}^2$) is about:

A) $4.5\ \text{Pa}$ B) $4.5\ \text{mPa}$ C) $4.5\ \mu\text{Pa}$ D) $4.5\ \text{nPa}$

Question 12

Solar radiation pressure acceleration on a spacecraft depends most strongly on its:

A) mass alone B) area alone C) area-to-mass ratio D) orbital period

Question 13

The Moon perturbs Earth satellites roughly twice as strongly as the Sun does because the tidal acceleration scales as:

A) $\mu_3 / d$ B) $\mu_3 / d^2$ C) $\mu_3 / d^3$ D) $\mu_3 \cdot d$

Question 14

The largest recurring station-keeping cost for a geostationary satellite fights:

A) atmospheric drag B) luni-solar inclination drift (north–south) C) solar radiation pressure D) J2 nodal regression

Question 15

Roughly how much delta-v per year does GEO north–south station-keeping require?

A) $\sim 0.5\ \text{m/s}$ B) $\sim 5\ \text{m/s}$ C) $\sim 50\ \text{m/s}$ D) $\sim 500\ \text{m/s}$

Question 16 (True/False, justify)

"J2 slowly lowers a satellite's altitude and circularizes its orbit." True or false? Justify in one sentence.

Question 17 (True/False, justify)

"A polar orbit experiences the maximum J2 nodal regression of any inclination." True or false? Explain.

Question 18 (True/False, justify)

"The air density at $400\ \text{km}$ is known precisely enough to predict a satellite's re-entry date years in advance." True or false? Say why.

Question 19 (Short answer)

Explain in one or two sentences why a sun-synchronous orbit must be tilted past $90^\circ$ (slightly retrograde) rather than exactly polar.

Question 20 (Short answer)

Write the two J2 secular rates ($\dot\Omega$ and $\dot\omega$), say in words what each one physically does to the orbit, and give the inclination that makes $\dot\omega = 0$.

Answer Key

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