A spacecraft executes a coplanar Hohmann transfer between two circular orbits of radii $r_1$ and $r_2$ (where $r_2 > r_1$). The semi-major axis $a_t$ of the elliptical transfer orbit is given by:

(A) \frac{r_1 + r_2}{2}

(B) \sqrt{r_1 r_2}

(C) \frac{2 r_1 r_2}{r_1 + r_2}

(D) r_2 - r_1

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Transfer Orbit Definition:** The Hohmann transfer ellipse is tangent to both the inner circular orbit at periapsis ($r_p = r_1$) and the outer circular orbit at apoapsis ($r_a = r_2$). 2. **Stage 2: Major Axis Relation:** $2a_t = r_a + r_p = r_1 + r_2$. 3. **Stage 3: Semi-major Axis:** $a_t = \frac{r_1 + r_2}{2}$. 4. **Stage 4: Selection:** Option A is correct.

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