# Interplanetary Trajectories Questions and Answers – J2 & J3 Effects

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

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 J2
d) Orthogonal coefficient

Explanation: The equatorial bulge of the Earth is modelled using the second order spherical harmonic function with zonal coefficient J2. This is about 1000 times larger than the other Jk coefficients.

3. How many orbital elements are affected by J2 perturbation?
a) 1
b) 2
c) 3
d) 4

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

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
RE is radius of earth = 6378 km
a = semi- major axis
e = eccentricity
μE is gravitational parameter of earth = 398600 km3/s2
i is orbit inclination.

5. For mars, only J2 perturbations are dominant.
a) True
b) False

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

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

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

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

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 – Jk $$\Big(\frac{R_E}{r}\Big)^2$$ Pk (sin⁡ Φ’)$$\Big]$$
b) U(r, Φ’) = $$\frac{\mu}{r} \Big[$$1 – J2 $$\Big(\frac{R_E}{r}\Big)^2$$ P2 (sin⁡ Φ’)$$\Big]$$
c) U(r, Φ’) = $$\frac{\mu}{r} \Big[$$1 – J2 $$\Big(\frac{R_E}{r}\Big)^2$$ P2 (cos Φ’)$$\Big]$$
d) U(r, Φ’) = $$\frac{\mu}{r} \Big[$$1 – Jk $$\Big(\frac{R_E}{r}\Big)^2$$ Pk (tan Φ’)$$\Big]$$

Explanation: The general geopotential function which depends on the radius and the altitude is given by: U(r, Φ’) = $$\frac{\mu}{r} \Big[$$1 – $$\Sigma_{k=2}^\infty$$Jk $$\Big(\frac{R_E}{r}\Big)^k$$ Pk (sin⁡ Φ’)$$\Big]$$ where Jk represents zonal harmonics, RE is the radius of the Earth and Pk is the Legendre polynomial.
Jk shows the buldges and dips on the surface of the Earth. But for the flattened earth, we use the J2 model as it is 1000 times larger than the other coefficients thus making the formulas as: U(r, Φ’) = $$\frac{\mu}{r} \Big[$$1 – J2 $$\Big(\frac{R_E}{r}\Big)^2$$ P2 (sin⁡ Φ’)$$\Big]$$.

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