AA 284a Advanced Rocket Propulsion Lecture 7b Gravitational Loss

Prepared by Arif Karabeyoglu

Department of Aeronautics and Astronautics Stanford University and Mechanical Engineering KOC University

Fall 2019 Stanford University AA284a Advanced Rocket Propulsion Gravity Loss Estimation Objective: • Derive a formula for the gravitational loss and gain intuition for this important loss mechanism Trajectory Equation: dV M = T − M gs dt

gs = g sin(γ ) dM dV = −I g − g dt sp o M s Real Delta V (velocity increment in gravitational field)

t ⎛ M i ⎞ ΔVr = V f −Vi = I sp go ln⎜ ⎟ − ∫ gsdt ⎝ M ⎠ o

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AA284a Advanced Rocket Propulsion Gravity Loss Estimation Specific energy: 1 s e V 2 g ds = + ∫ s 2 o Here s is the distance along the trajectory.

de =VdV + gsds

Trajectory Eqn.: dM VdV = −I sp goV − gsVdt M Eliminate dt dM ds VdV I g V g ds = V = − sp o − s dt M Stanford University 3 Karabeyoglu

AA284a Advanced Rocket Propulsion Gravity Loss Estimation Combine: dM de = −I g V sp o M

Ideal and real velocity increase

dM dM de = −I g V de = −I g V id sp o id M r sp o r M Gravity loss: dM dG ≡ deid − der = −Ispgo (Vid −Vr ) M

t dM dG = −I sp go gsdt M ∫ 0 Stanford University 4 Karabeyoglu

AA284a Advanced Rocket Propulsion Gravity Loss Estimation Use of chain rule: t ⎡ ⎛ M ⎞ ⎤ ⎛ M ⎞ dG I g d ⎢ln⎜ ⎟ g dt⎥ I g ln⎜ ⎟g dt = − sp o ⎜ ⎟ s + sp o ⎜ ⎟ s ⎢ M i ∫ ⎥ M i ⎣ ⎝ ⎠ 0 ⎦ ⎝ ⎠ Integrate over burn time:

t t ⎛ M ⎞ b b ⎛ M ⎞ G I g ln b g dt I g ln g dt = − sp o ⎜ ⎟∫ s + sp o ∫ ⎜ ⎟ s ⎝ Mi ⎠ 0 0 ⎝ Mi ⎠

t b ⎡ ⎛ M ⎞ ⎛ M ⎞⎤ G = I g ln⎜ ⎟ + ln⎜ i ⎟ g dt sp o ∫ ⎢ ⎜ M ⎟ ⎜ M ⎟⎥ s o ⎣ ⎝ i ⎠ ⎝ b ⎠⎦

tb ⎛ M ⎞ G I g ln⎜ ⎟g dt = sp o ⎜ ⎟ s ∫ M b o ⎝ ⎠

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AA284a Advanced Rocket Propulsion Gravity Loss Estimation Reintroduce time: m! = − dM dt

Mb g ⎛ M ⎞ G I g s ln⎜ ⎟dM = − sp o ⎜ ⎟ ∫ m! ⎝ M b ⎠ Mi For constant gs/m

M b ⎪⎧ M i ⎡ ⎛ M i ⎞ ⎤⎪⎫ G = I sp go gs ⎨1+ ⎢ln⎜ ⎟ −1⎥⎬ m M ⎜ M ⎟ ! ⎩⎪ b ⎣⎢ ⎝ b ⎠ ⎦⎥⎭⎪ Constant acceleration

2 T m! Ispgo 1 g ⎡ ⎛ M ⎞⎤ a = = G = s I g ln⎜ i ⎟ M M ⎢ sp o ⎜ ⎟⎥ 2 a ⎣ ⎝ M b ⎠⎦

G g ⎛ M ⎞ = s ln2 ⎜ i ⎟ 1 2V 2 a ⎜ M ⎟ e ⎝ b ⎠ Stanford University 6 Karabeyoglu

AA284a Advanced Rocket Propulsion Gravity Loss Estimation Conclusions: • The gravity loss decreases with increasing mass flow rate or acceleration (decreasing burn time) • In the coast phase the gravity loss is zero.

• Gravity loss decreases with decreasing gs (or with decreasing average trajectory angle) • The coast phase reduces the overall gravity loss by decreasing the combined burn time for all stages and also by reducing the average trajectory angle for the upper stages. Note that during the coast phase there is no gravity loss but the trajectory angle continues to drop (gravity turn).

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AA284a Advanced Rocket Propulsion Gravity Loss Estimation • The formula for the specific energy increase given in slide 4 is instrumental in showing the relative importance of Isp for upper stages. This equation can be rearranged to express the increase in specific energy for unit mass of propellant expelled from the rocket, namely the sensitivity coefficient . de I g V σ = = sp o dM M • Note that for an efficient rocket system the sensitivity coefficient should be maximized. Since the upper stages of a launch vehicle are lighter and move faster, a unit increase in the Isp value results in a larger increase in the sensitivity parameter for the upper stages. • The loss described in this write up is not limited to the gravitational forces. As can be deduced from gravity loss formula, any force that limits the velocity built up of the rocket (i.e. any force that acts over a finite period against the direction of the thrust force) will result in a loss of useful energy. Some examples of such forces are • Drag force • Static firing of a rocket. The force exerted by the thrust stand results in a delta V loss equal to the delivered delta V by the rocket. Stanford University 8 Karabeyoglu