Lesson 10: Orbital Transfers I

10/6/2016 Robin Wordsworth ES 160: Space Science and Engineering: Theory and Applicaons Objecves

• Understand the connuous and impulsive thrust approximaons • Revisit specific energy and the vis viva equaon • Introduce the Hohmann transfer orbit ONLY IN THE MOVIES! Key Objecves in Orbital Transfers

1. Minimize energy cost 2. Minimize propellant cost 3. Minimize transit me Key Objecves in Orbital Transfers

1. Minimize energy cost 2. Minimize propellant cost (delta-v) 3. Minimize transit me Connuous thrust transfers

• Solar sails • Ion propulsion • Theorecal analysis requires integraon over many orbits – can become very tricky

Staehle R., et al. (2013) Impulsive Thrust Approximaon

• Transfer calculaons are greatly simplified if we can treat burn me as << orbital period • Usually good for chemical rockets. E.g. for 500 km circular LEO, T = 1.6 h. Burn me of 60 s à 1% of total. • For interplanetary trajectories, approximaon is even beer

Prussing & Conway, Orbital Mechanics Impulsive Thrust Approximaon

• Transfer calculaons are greatly simplified if we can treat burn me as << orbital period Δv • Usually good for chemical v rockets. E.g. for 500 km old circular LEO, T = 1.6 h. Burn v me of 60 s à 1% of total. new • For interplanetary trajectories, approximaon is even beer The general transfer orbit

• Impulse gives instantaneous (arbitrary) change in velocity • Immediately aer impulse, spacecra orbit is changed • Transfer orbits get you from original to desired orbit

Prussing & Conway, Orbital Mechanics The general circle-to-circle transfer orbit Basic constraints: rp r1 p r1(1 + e)   r r p r2(1 e) a 2 2

ALLOWED 1.5 r 2 = r a 1 r p = ALLOWED

eccentricity e r 0.5 1 FORBIDDEN

FORBIDDEN FORBIDDEN 0 0 0.5 1 1.5 2 2.5 3 parameter p The Hohmann Transfer

• Conjectured to be the minimum-fuel two- impulse transfer orbit by Walter Hohmann in 1925 • Both conceptually simple and praccally important • For minimum fuel proof, see e.g. Prussing & Conway, Orbital Mechanics, p. 125-126

hps://www.astro-shop.com/Die-Erreichbarkeit-der-Himmelskoerper.html Pisacane, Fundamentals of Space Systems The general circle-to-circle transfer orbit Basic constraints: rp r1 p r1(1 + e)   r r p r2(1 e) a 2 2

1.5 HYPERBOLAS Hohmann r 2 transfer = r a 1 PARABOLAS r p = ELLIPSES

eccentricity e r 0.5 1 FORBIDDEN

FORBIDDEN FORBIDDEN 0 0 0.5 1 1.5 2 2.5 3 parameter p An Example: Earth-Venus transfer

• We want to travel to Venus in a Hohmann transfer. Assume we have already calculated launch window. • What delta-v do we need for the transfer? • Neglect the gravitaonal aracon of Earth and VENUS Venus… for now

EARTH An Example: Earth-Venus transfer

v 2 R r /r A =1 ⌘ 2 1 v2 r1+R v 2R VENUS B ΔvB = 1 r2 v1 r1+R Note direcon is opposite from example in the notes! r1 • R = 1 AU / 0.723 AU = 1.383

• v1 = 35.0 km/s

• v2 = 28.8 km/s

• ΔvA = 2.49 km/s

• ΔvB = 2.71 km/s

ΔvA EARTH Summary

• Lowest energy (Hohmann) transfer between two circular orbits is an ellipse with periapsis and apoapsis that matches the two radii • Required speed change (delta-v) can be derived from vis viva equaon (or equivalently, just conservaon of energy)