GATE 2022 AE – Question 46
Consider a high Earth-orbiting satellite of angular momentum per unit mass hሬ⃗ and eccentricity e. The mass of the Earth is M and G is the universal gravitational constant. The distance between the satellite’s center of mass and the Earth’s center of mass is r, the true anomaly is θ, and the phase angle is zero. Which of the following statements is/are true?
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Correct answer: (C) Specific angular momentum is conserved.; (D) Potential plus kinetic energy is conserved.
Explanation
**Two-body (Kepler) orbit under central gravity.**
- **Specific angular momentum $\vec h=\vec r\times\vec v$ is conserved** because gravity is a central force (no torque about the Earth's centre). So **C is true**. ✓
- **Mechanical energy (kinetic + potential) per unit mass is conserved** because gravity is a conservative force and nothing else acts. So **D is true**. ✓
**The orbit equation** (conic section in polar form) is
$$r=\frac{h^2/GM}{1+e\cos\theta}=\frac{|h|^2}{GM\,(1+e\cos\theta)}.$$
In options A and B, $G$ and $M$ appear in the numerator instead of the denominator, and A even has $|h|$ to the first power. So **A and B are false**.
Answer **C and D**.