Chapter 6: Maneuvering in Space

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 (km2/s2) μ = gravitational parameter = 3.986 x 105 (km3/s2) 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 1st orbit to 2nd orbit.

7. (change in velocity) ΔV = |Vselected – Vpresent| -------------------------------------------------------------------------- (change in velocity that takes spacecraft from orbit 1 into transfer orbit) (km/s) ΔV1 = |Vtransfer at orbit 1 – Vorbit 1| Vtransfer at orbit 1 = velocity in the transfer orbit 1 radius (km/s) Vorbit 1 = velocity orbit 1 (km/s) -------------------------------------------------------------------------- (change in velocity that takes spacecraft from transfer orbit into orbit 2) (km/s) ΔV2 = |Vorbit 1 – Vtransfer at orbit 2|

8. (Total velocity change needed to for the transfer) (km/s)ΔVtotal = ΔV1 + ΔV2 orbit2 orbit1 Transfer orbit

9. Mass Calculations(pay attention!!) To compute ΔVtotal , we use the energy equations from orbital mechanics. ε = V2 – μ 2 R ε = specific mechanical energy (km2/s2) V = magnitude of the spacecraft’s velocity vector (km/s) μ = gravitational parameter = 3.986 x 105 (km3/s2) for Earth R = magnitude of the spacecraft’s position vector (km) ε = -μ 2a ε = specific mechanical energy (km2/s2) μ = gravitational parameter = 3.986 x 105 (km3/s2) for Earth a = semi-major axis (km)

10. Review steps in transfer process: • Step 1: ΔV1 takes a spacecraft from orbit 1 and puts it into the transfer orbit. • Step 2: ΔV2 puts the spacecraft into the orbit 2 from the transfer orbit. To solve for ΔVs, find the energy in each orbit: 2atransfer = Rorbit1 + Rorbit2 Using alternate equation for specific mechanical energy: eorbit1 =_ -μ _ , eorbit2 =_ -μ _etransfer =_ -μ _ 2aorbit1 2aorbit1 2atransfer

11. Calculate velocities: Vorbit1 =sqrt ( 2( _ μ_+ eorbit1) ) Rorbit1 Vorbit2 =sqrt ( 2( _ μ_+ eorbit2) ) Rorbit2 Vtransfer at orbit1 =sqrt ( 2( _ μ_+ etransfer) ) Rorbit1 Vtransfer at orbit2 =sqrt ( 2( _ μ_+ etransfer) ) Rorbit2

12. ΔV1 = |Vtransfer at orbit 1 – Vorbit 1| ΔV2 = |Vorbit 1 – Vtransfer at orbit 2| ΔVtotal = ΔV1 + ΔV2 ---------------------------------------------------- (Transfer orbit’s time of flight (TOF) is half of the period) TOF = P = p x sqrt( a3transfer) 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 105 (km3/s2) for Earth