1. Chapter 6: Maneuvering in Space By: Antonio Batiste

2. 6.1: Hohmann Transfers Theorized in 1925 by German engineer Walter Hohmann. Concluded to be the most fuel-efficient way to maneuver in space. Uses an elliptical transfer orbit tangent to the initial and final orbits.

3. We limit Hohmann Transfers: Orbits in the same plane (coplanar orbits). Orbits with their major axes (line of apsides) aligned (co-apsidal orbits) or cirular orbits. Instantaneous velocity changes (ΔVs) tangent to the initial and final orbits (make the Hohmann Transfer the most efficient transfer).

4. Limiting Hohmann Transfers in more detail: Co-apsidal orbits take their name because two elliptical orbits have their major axes (line of apsis) aligned with one another. Velocity changes are “instantaneous” because we assume that the time the engine fires is very short compared to the Hohmann Transfer time of flight. NOTE: Impulsive Burn: a 2-5 minute burn; nearly instantaneous.

5. Whenever we add or subtract velocity, we change the orbit’s specific mechanical energy (ε) ε = - μ 2a ε = specific mechanical energy (km 2 /s 2 ) μ = gravitational parameter = 3.986 x 10 5 (km 3 /s 2 ) for Earth  a = semi-major axis (km)

6. To move spacecraft to higher orbit we have to increase the semi-major axis (adding energy to the orbit) by increasing velocity. To move spacecraft to lower orbit, we have to decrease the semi-major axis (and the energy) by decreasing the velocity. Transfer Orbit – junction; orbit needed to get to 1 st orbit to 2 nd orbit.

7. (change in velocity) ΔV = |V selected – V present | -------------------------------------------------------------------------- (change in velocity that takes spacecraft from orbit 1 into transfer orbit) (km/s) ΔV 1 = |V transfer at orbit 1 – V orbit 1 | V transfer at orbit 1 = velocity in the transfer orbit 1 radius (km/s) V orbit 1 = velocity orbit 1 (km/s) -------------------------------------------------------------------------- (change in velocity that takes spacecraft from transfer orbit into orbit 2) (km/s) ΔV 2 = |V orbit 1 – V transfer at orbit 2 |

8. (Total velocity change needed to for the transfer) (km/s) ΔV total = Δ V 1 + ΔV 2 orbit 1 Transfer orbit orbit 2

9. Mass Calculations (pay attention!!) To compute ΔV total, we use the energy equations from orbital mechanics. ε = V 2 – μ 2 R ε = specific mechanical energy (km 2 /s 2 ) V = magnitude of the spacecraft’s velocity vector (km/s) μ = gravitational parameter = 3.986 x 10 5 (km 3 /s 2 ) for Earth R = magnitude of the spacecraft’s position vector (km) ε = - μ 2a ε = specific mechanical energy (km 2 /s 2 ) μ = gravitational parameter = 3.986 x 10 5 (km 3 /s 2 ) for Earth  a = semi-major axis (km)

10. Review steps in transfer process: Step 1: ΔV 1 takes a spacecraft from orbit 1 and puts it into the transfer orbit. Step 2: ΔV 2 puts the spacecraft into the orbit 2 from the transfer orbit. To solve for ΔVs, find the energy in each orbit: 2a transfer = R orbit1 + R orbit2 Using alternate equation for specific mechanical energy:  orbit1 = _ - μ _,  orbit2 = _ - μ _  transfer = _ - μ _ 2a orbit1 2a orbit1 2a transfer

11. Calculate velocities: V orbit1 = sqrt ( 2( _ μ_+  orbit1 ) ) R orbit1 V orbit2 = sqrt ( 2( _ μ_+  orbit2 ) ) R orbit2 V transfer at orbit1 = sqrt ( 2( _ μ_+  transfer ) ) R orbit1 V transfer at orbit2 = sqrt ( 2( _ μ_+  transfer ) ) R orbit2

12. ΔV 1 = |V transfer at orbit 1 – V orbit 1 | ΔV 2 = |V orbit 1 – V transfer at orbit 2 | ΔV total = Δ V 1 + ΔV 2 ---------------------------------------------------- (Transfer orbit ’ s time of flight (TOF) is half of the period) TOF = P =  x sqrt( a 3 transfer ) 2 μ TOF = spacecraft’s time of flight (s) P = orbital period (s) a = semi-major axis of the transfer orbit (km) μ = gravitational parameter = 3.986 x 10 5 (km 3 /s 2 ) for Earth