A spacecraft transfers between two coplanar circular orbits of radius $r_1$ and $r_2$ where $r_2 > r_1$. Which of the following statements regarding the fuel efficiency of a 3-impulse bi-elliptic transfer compared to a 2-impulse Hohmann transfer are **CORRECT**?

(A) For radius ratios $\frac{r_2}{r_1} < 11.94$, the Hohmann transfer always requires less total characteristic velocity change $\Delta V$ than any bi-elliptic transfer

(B) For radius ratios $\frac{r_2}{r_1} > 15.58$, a bi-elliptic transfer with sufficiently large intermediate apocenter radius $r_i$ is strictly more fuel-efficient than a Hohmann transfer

(C) A bi-elliptic transfer requires three impulsive velocity burns whereas a Hohmann transfer requires two

(D) A bi-elliptic transfer always requires less total flight time than a Hohmann transfer between the same two orbits

Answer Key:

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### 1. Strategy & Theoretical Principles Comparison between Hohmann (2-burn) and Bi-Elliptic (3-burn) transfers: 1. **Radius Ratio Regimes:** - For $\frac{r_2}{r_1} < 11.938$: The Hohmann transfer is strictly optimal for all intermediate radii $r_i$. - For $11.938 < \frac{r_2}{r_1} < 15.58$: Whether bi-elliptic is superior depends on the intermediate radius $r_i$. - For $\frac{r_2}{r_1} > 15.58$: A bi-elliptic transfer with $r_i > r_2$ is always more fuel-efficient (lower $\Delta V_{\text{total}}$) than the Hohmann transfer. 2. **Burn Count & Flight Time:** - Hohmann uses 2 impulsive burns (LEO to transfer, transfer to GEO). - Bi-elliptic uses 3 impulsive burns (burn 1 to reach outer apocenter $r_i$, burn 2 at $r_i$ to raise pericenter to $r_2$, burn 3 at $r_2$ to circularize). - Because bi-elliptic loops out to a distant apocenter $r_i > r_2$, its total transfer flight time is substantially *longer* than the Hohmann transfer time. ### 2. Statement Evaluation - **Statement A:** **CORRECT.** $R^* = 11.938$ is the strict lower boundary for Hohmann superiority. - **Statement B:** **CORRECT.** Above $15.58$, bi-elliptic is always more fuel-efficient. - **Statement C:** **CORRECT.** Bi-elliptic is a 3-impulse maneuver. - **Statement D:** **INCORRECT.** Bi-elliptic transfer takes significantly longer flight time. ### 3. Exam Trap & Standard Reference - **Trap Alert:** The trade-off for bi-elliptic fuel savings is drastically longer transit time. - **Standard Reference:** Curtis, H.D., *Orbital Mechanics for Engineering Students*, Chapter 6.

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