GATE AE 2026 • Q55 NAT • 2 Marks Flight Mechanics & Space Dynamics

Flight Mechanics & Space Dynamics

Space Dynamics & Orbital Mechanics

Subtopic: Kepler's Laws & Two-Body Problem

Question Statement:

An earth satellite moves in an elliptical orbit with a perigee altitude of 300 km and an apogee altitude of 3000 km. Assume that the radius of the earth is 6378 km. The eccentricity of the orbit is __________ (rounded off to three decimal places).

Official Answer Key:

0.168

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2026 Q.55)** **1. Intuitive Physical Setup & Given Parameters**: An Earth satellite moves in an elliptical orbit: - Earth radius: $R_E = 6378\text{ km}$ - Perigee altitude: $h_p = 300\text{ km}$ - Apogee altitude: $h_a = 3000\text{ km}$ We need to determine the eccentricity $e$ of the orbit, rounded off to three decimal places. **2. Step-by-Step Orbital Mechanics Derivation**: 1. **Perigee and Apogee Radii**: The orbital radii measured from Earth's geocenter are: r_p = R_E + h_p = 6378 + 300 = 6678\text{ km} r_a = R_E + h_a = 6378 + 3000 = 9378\text{ km} 2. **Formula for Orbital Eccentricity**: For any Keplerian elliptical orbit: r_p = a(1 - e), \quad r_a = a(1 + e) Dividing the difference by the sum: e = \frac{r_a - r_p}{r_a + r_p} 3. **Calculate Eccentricity**: r_a - r_p = 9378 - 6678 = 2700\text{ km} r_a + r_p = 9378 + 6678 = 16056\text{ km} e = \frac{2700}{16056} = \frac{225}{1338} \approx 0.168161 **3. Comprehensive Options & Distractor Analysis**: - **Evaluation Criteria**: Numerical Answer Type (NAT). - **Exact Analytical Result**: $e \approx 0.16816$. - **Rounding Requirement**: Three decimal places $\implies \mathbf{0.168}$. **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **0.168** (Acceptable Range: **0.165 to 0.170**) - **Exam Takeaway**: Always add Earth's radius $R_E$ to convert altitudes to geocentric radii before computing $e = \frac{r_a - r_p}{r_a + r_p}$.
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