This set of Spaceflight Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Interplanetary Trajectories – J2 & J3 Effects”.

1. Which of these perturbing forces is a result of earth’s oblateness?

a) J2 Perturbation

b) J3 perturbation

c) J2 and J3 perturbation

d) J5 perturbation

View Answer

Explanation: Earth’s not a complete spherical body, instead it’s equator is 21 km larger than radius at the poles. This oblateness effect causes perturbing forces on object orbiting the earth. This perturbing force based on oblate Earth is known as J2 Perturbations.

2. Which of the second order harmonics functions is used to model the bulge of the Earth?

a) Tesseral coefficient

b) Sectoral coefficient

c) Zonal coefficient J_{2}

d) Orthogonal coefficient

View Answer

Explanation: The equatorial bulge of the Earth is modelled using the second order spherical harmonic function with zonal coefficient J

_{2}. This is about 1000 times larger than the other J

_{k}coefficients.

3. How many orbital elements are affected by J2 perturbation?

a) 1

b) 2

c) 3

d) 4

View Answer

Explanation: Out of 6 orbital elements that define the orbit around the earth, 2 are affected by the J2 perturbations. These are Right Ascension of the Ascending Node Ω and Argument of Perigee ω.

4. What is the formula used to compute the change in Right Ascension of the Ascending Node over time due to J2 perturbations?

a) \(\frac{d\bar{\Omega}}{dt} = \frac{-3nJ_2}{2} \Big(\frac{R_E}{a(1-e^2)}\Big)^2 \sqrt{\frac{\mu_E}{a^3}}\) cos i

b) \(\frac{d\bar{\Omega}}{dt} = \frac{-3nJ_2}{2} \Big(\frac{R_E}{a}\Big)^2\) cos i

c) \(\frac{d\bar{\Omega}}{dt} = \frac{-3nJ_2}{2} \Big(\frac{R_E}{a(1-e^2)}\Big)^2\)

d) \(\frac{d\bar{\Omega}}{dt} = \frac{-3nJ_2}{2} \Big(\frac{R_E}{a(1-e^2)}\Big)^2 \sqrt{\frac{\mu_E}{a^3}}\) sin i

View Answer

Explanation: J2 perturbation plays a huge role in changing the value of RAAN over time. The formula to compute this change is given by:

\(\frac{d\bar{\Omega}}{dt} = \frac{-3nJ_2}{2} \Big(\frac{R_E}{a(1-e^2)}\Big)^2 \sqrt{\frac{\mu_E}{a^3}}\) cos i

Where, \(\frac{d\bar{\Omega}}{dt}\) is nodal precession

J2 = constant = 1.08262668 × 10

^{-3}

R

_{E}is radius of earth = 6378 km

a = semi- major axis

e = eccentricity

μ

_{E}is gravitational parameter of earth = 398600 km

^{3}/s

^{2}

i is orbit inclination.

5. For mars, only J2 perturbations are dominant.

a) True

b) False

View Answer

Explanation: When the orbit perturbations are considered in case of Earth, only J2 effects are considered since it is dominant over other nodal regressions. Meanwhile, in Mars both J2 and J3 perturbations are dominant. J2 = 1.955 × 10

^{-3}and J3 = 3.14498 × 10

^{-5}which are important for the determination of frozen orbits around Mars.

6. J3 perturbation is zero for Mercury and Venus.

a) True

b) False

View Answer

Explanation: The term J3 is a zonal (axial symmetry) term that expresses the effects of asymmetry between the hemispheres of the north and the south. J3 is zero or very small for Mercury and Venus, which are essentially spherical.

7. For while celestial bodies are J3 perturbations negligible?

a) Comets

b) Asteroid

c) Giant planets

d) Sun

View Answer

Explanation: J3 is zero for the giant planets since the plasticity of these planets produces only zonal coefficients of J2n (symmetry relative to the equatorial plane or symmetry between north and south).

8. Which of these orbital elements are affected by the J3 perturbation?

a) Right Ascension

b) Argument of perigee

c) Eccentricity

d) Inclination

View Answer

Explanation: Jr perturbation has an impact on the orbital inclination. Unlike J2 perturbation, it has very weak effect on the right of ascension and argument of perigee. The disturbance effects of J3 perturbations on the argument of perigee, right ascension and eccentricity are weakened when the eccentricity increases such as for the frozen orbits (Molniya or Tundra orbit).

9. For a Low Earth orbiting satellite, which of these perturbations cannot be overlooked?

a) J2

b) J3

c) J2 and J3

d) J5

View Answer

Explanation: Usually J2 perturbations are more dominant compared to J3 perturbation for satellites orbiting the earth. But, J3 perturbation on LEO satellites cannot be overlooked.

10. What is the formula of the geopotential function that is used to model the oblate Earth?

a) U(r, Φ’) = \(\frac{\mu}{r} \Big[\)1 – J_{k} \(\Big(\frac{R_E}{r}\Big)^2\) P_{k} (sin Φ’)\(\Big]\)

b) U(r, Φ’) = \(\frac{\mu}{r} \Big[\)1 – J_{2} \(\Big(\frac{R_E}{r}\Big)^2\) P_{2} (sin Φ’)\(\Big]\)

c) U(r, Φ’) = \(\frac{\mu}{r} \Big[\)1 – J_{2} \(\Big(\frac{R_E}{r}\Big)^2\) P_{2} (cos Φ’)\(\Big]\)

d) U(r, Φ’) = \(\frac{\mu}{r} \Big[\)1 – J_{k} \(\Big(\frac{R_E}{r}\Big)^2\) P_{k} (tan Φ’)\(\Big]\)

View Answer

Explanation: The general geopotential function which depends on the radius and the altitude is given by: U(r, Φ’) = \(\frac{\mu}{r} \Big[\)1 – \(\Sigma_{k=2}^\infty\)J

_{k}\(\Big(\frac{R_E}{r}\Big)^k\) P

_{k}(sin Φ’)\(\Big]\) where J

_{k}represents zonal harmonics, R

_{E}is the radius of the Earth and P

_{k}is the Legendre polynomial.

J

_{k}shows the buldges and dips on the surface of the Earth. But for the flattened earth, we use the J

_{2}model as it is 1000 times larger than the other coefficients thus making the formulas as: U(r, Φ’) = \(\frac{\mu}{r} \Big[\)1 – J

_{2}\(\Big(\frac{R_E}{r}\Big)^2\) P

_{2}(sin Φ’)\(\Big]\).

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