Chapter 6:
Maneuvering in Space
By: Antonio Batiste
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.
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).
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.
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)
• 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.
(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|
(Total velocity change needed to for the transfer) (km/s)
ΔVtotal = ΔV1 + ΔV2
orbit2
orbit1
Transfer orbit
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)
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
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
)
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
)