A satellite of mass $m = 500\text{ kg}$ is in an elliptical orbit around Earth with a perigee radius $r_p = 7{,}000\text{ km}$ and apogee radius $r_a = 9{,}000\text{ km}$. Earth's gravitational parameter is $\mu = 398{,}600\text{ km}^3/\text{s}^2$. The specific mechanical energy $\mathcal{E}$ (in $\text{km}^2/\text{s}^2$) of the orbit is:

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### 1. Strategy & Theoretical Principles From orbital mechanics energy conservation, the specific mechanical energy $\mathcal{E}$ of any Keplerian 2-body orbit depends solely on the semi-major axis $a$: $$\mathcal{E} = -\frac{\mu}{2a}$$ For an elliptical orbit with perigee $r_p$ and apogee $r_a$: $$a = \frac{r_p + r_a}{2}$$ ### 2. Analytical Formulation & Calculation Given: - $r_p = 7{,}000\text{ km}$ - $r_a = 9{,}000\text{ km}$ - $\mu = 398{,}600\text{ km}^3/\text{s}^2$ 1. Semi-major axis: $$a = \frac{7000 + 9000}{2} = \frac{16{,}000}{2} = 8{,}000\text{ km}$$ 2. Specific energy: $$\mathcal{E} = -\frac{398{,}600}{2 \times 8{,}000} = -\frac{398{,}600}{16{,}000} = -24.9125\text{ km}^2/\text{s}^2 \approx -24.91\text{ km}^2/\text{s}^2$$ ### 3. Distractor Forensics - Candidate omitting the negative sign for bound orbits reports $+24.91$, which would represent an unbound hyperbolic orbit. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Bound closed orbits (circular, elliptical) always have negative specific energy ($\mathcal{E} < 0$). - **Standard Reference:** Curtis, H.D., *Orbital Mechanics for Engineering Students*, Chapter 2.

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