Spaceflight Mechanics Questions and Answers – Perturbations

This set of Spaceflight Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Perturbations”.

1. What is the Keplerian orbit that at a specific time having the same orbital elements of perturbed trajectory called?
a) Perturbed orbit
b) Osculating orbit
c) Keplerian orbit
d) Non Keplerian orbit

Explanation: The Keplerian orbit that at a specific time has the same orbital elements of the perturbed trajectory is called osculating orbit.The osculating orbital elements of a spacecraft are the parameters of its osculating conic section, which varies with time.

2. Which of these is not the cause of orbital perturbations?
a) Aerodynamic drag
b) Atmospheric drag
c) Asphericity of Earth
d) Climate changes

Explanation: Perturbations are disturbances causing the object to deviate from the original orbit. The main reasons for perturbations are: aerodynamic drag, asphericity of earth, atmospheric drag, radiation pressure, solar wind pressure and electromagnetic effect. Climate change on the other hand is a result of orbital perturbations.

3. The object in the orbit follows Keplerian motion due to the orbital perturbations.
a) True
b) False

Explanation: When the object or a satellite is perturbed due to any additional forces such as gravitational force, atmospheric drag, the two-body problem is not applicable to the object. It follows non-Keplerian motion.

4. Which of these equations show the modified two-body equation with perturbing vector incorporated in it?
a) $$\ddot{\vec{r}}$$ = $$\frac{-\mu}{r^3} \vec{r} + \vec{a}_p$$
b) $$\ddot{\vec{r}}$$ = $$\frac{-\mu}{r^3} \vec{r} – \vec{a}_p$$
c) $$\ddot{\vec{r}}$$ = $$\frac{-\mu}{r^2} \vec{r} + \vec{a}_p$$
d) $$\ddot{\vec{r}}$$ = $$\frac{-\mu}{r^2} \vec{r} – \vec{a}_p$$

Explanation: The governing two-body problem without any perturbation is given by:
$$\ddot{\vec{r}}$$ + $$\frac{-\mu}{r^3} \vec{r}$$ = 0
When there are external forces that alter the orbit, we have to add the perturbing vector thus making it:
$$\ddot{\vec{r}}$$ = $$\frac{-\mu}{r^3} \vec{r} + \vec{a}_p$$
Here, $$\vec{a}_p$$ is the accelerating perturbing vector which acts on the object in the orbit. This is computed by adding all the external forces such as atmospheric drag, aerodynamic drag, solar radiation etc and then dividing it with mass of the object circling in the orbit. This vector is expressed in ECI frame of reference.

5. Which of these orbital perturbations is not a result of the Earth’s oblateness?
a) Torque at the center of Earth
b) Rotation of line of apsides
c) Change in object’s attitude dynamics
d) Nodes moves westward for direct orbits

Explanation: The Earth is not a perfectly spherical body, rather it is oblate with poles that are flattened. Due to this there is a density variation inside the volume. The excess mass near the equator causes slight torque on the satellite about the center of the Earth. The torque causes the orbit plane to precess, and the line of the nodes moves westward for direct orbits and eastward for retrograde satellites. Earth’s oblateness also causes a rotation of the line of apsides.

6. What is the secular drift rate in Ω caused by lunar gravity for the satellite orbiting the Earth at an inclination of 20 degrees with its semi major axis of 8000 km?
a) -0.00026468 ($$\frac{deg}{day}$$)
b) -0.002847 ($$\frac{deg}{day}$$)
c) -0.0001874 ($$\frac{deg}{day}$$)
d) -0.00002174 ($$\frac{deg}{day}$$)

Explanation: Given, i = 20°, a = 8000 km
The secular drift in Ω due to moon’s gravity on the satellite orbiting Earth is given by:
$$\dot{\bar{\Omega}}_m$$ = $$\frac{-0.00338}{N_{rev}}$$ cos ⁡i ($$\frac{deg}{day}$$)
Where Nrev is the number of orbital revolutions per day
To calculate the Nrev, we fist need to find the orbital time-period:
Tperiod = $$\frac{2\pi}{\sqrt{\mu}} a^{\frac{3}{2}} = \frac{2\pi}{\sqrt{398,600}} 8,000^{\frac{3}{2}}$$ = 7121 seconds = 1.978 hours
Number of revolutions per day = $$\frac{24 \,hours}{1.978 \,hours}$$ = 12.133 ≅ 12 revolutions
$$\dot{\bar{\Omega}}_m$$ = $$\frac{-0.00338}{12}$$ cos⁡ 20° = -0.00026468 ($$\frac{deg}{day}$$)

7. What is the value of solar radiation pressure for a satellite orbiting Earth if the solar intensity is 1,361 W/m2?
a) 2.21 × 10-6 N/m2
b) 6.36 × 10-6 N/m2
c) 4.54 × 10-6 N/m2
d) 3 × 10-6 N/m2

Explanation: Given, Is = 1,361 W/m2
When the satellite orbits Earth, it experiences solar radiation pressure. The solar intensity is already mentioned which is, Is = 1,361 W/m2. Thus, the solar radiation pressure PSRP is calculated by dividing the solar intensity with speed of light.
PSRP = $$\frac{I_s}{c} = \frac{1,361}{3 × 10^8}$$ = 4.54 × 10-6 $$\frac{N}{m^2}$$ or Pascals

8. In which layer of the atmosphere is the satellite likely to be perturbed by the electromagnetic effect?
a) Stratosphere
b) Exosphere
c) Ionosphere
d) Troposphere

Explanation: A satellite may acquire electrostatic charges in the ionized high-altitude atmosphere. Interactions with the Earth’s magnetic field are possible thus causing perturbations and altering satellite’s attitude dynamics.

9. Which of these external forces causes perturbations due to the presence of atmosphere?
a) Solar Pressure
b) Aerodynamic Pressure
c) Magnetic Pressure
d) Gravitational Variation

Explanation: The pressure due to the atmosphere affects the satellite even though the space seems like vacuum. If the center of pressure of the object does not coincide with the center of mass, pressure acts on the body resulting in torque due to the presence of the atmosphere.

10. Which of these is the cause of the rotation of the line of apsides?
a) Oblateness of Earth
b) Inclination of Earth
d) Magnetic pressure

Explanation: Since the equator of the earth bulges out whereas the poles are flattened at both the hemispheres, this causes the satellite to drift due to the regression of nodes causing rotation of the line of apsides.

11. Which of these methods was used to predict Halley’s comet return?
a) Cowell’s method
b) Encke’s method
c) Euler’s method
d) Kepler’s method

Explanation: Cowell’s method was used to predict the exact time of the approach of the return of Haylley’s comet in 1910. This method integrates sum of accelerations numerically.
$$\frac{d^2 \vec{r}}{dt^2} + \frac{\mu}{r^3} \vec{r} = \vec{a_p}$$
Where, $$\vec{a_p}$$ is sum of the accelerations die to perturbations.

12. Which of these Keplerian elements is slow variable?
a) θ
b) tp
c) M
d) ω

Explanation: There are six Keplerian elements as follows: a, e, tp, Ω, ω, i. Out of these, there are few elements a, e, Ω, ω, i which are unaffected by the perturbations. These are known as ‘slow variables’.

13. ‘Fast variables’ are those variables that changes with perturbations.
a) True
b) False

Explanation: There are few elements that vary when there are perturbations. These are true anomaly θ, Mean anomaly M and time period tp. These are referred as ‘fast variables’.

More MCQs on Perturbations:

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