Orbital Motion

2/28/19

Seminar 3 Precursors to Space Flight Orbital Motion FRS 148, Princeton University Robert Stengel Prophets with Some Honor The Human Seed and Social Soil: Rocketry and Revolution Orbital Motion Energy and Momentum Orbital Elements

Precursors to Space Flight: Rocket, Missiles, and Men in Space, Ch 4 (ER) … the Heavens and the Earth, Introduction, Ch 1 Orbital Motion: Understanding Space, Ch 5

Prophets with Some Honor

Willy Ley, 1906-1969 2

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Konstantin Tsiolkovsky (1857-1935)

• Russian “father of spaceflight” • High school teacher who wrote of rocket-propelled vehicles • “The rocket equation” (1897): vehicle speed change depends on – Rocket exhaust velocity – Ratio of vehicle’s full-to-empty mass m Δv = v log initial exhaust e m final 3

Robert Esnault-Pelterie (1881-1957)

• Airplane design & flight, 1907 • Rocket equation • Atomic propulsion proposal • Ballistic missile proposal • Interplanetary flight studies

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Pangaea and Eusthenopteron

Walter McDougall 5 1946-

Industrial Revolution

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Electric Power Storage, Generation, and Transmission

Industrial Revolution and Government Science

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Long-Distance Communication

Government, Technology, and War

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Introduction, …the Heavens and the Earth

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Introduction, …the Heavens and the Earth

Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution

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Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution

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Ch. 1, The Human Seed and Social Soil: Rocketry and Revolution

Early 20th Century Rocket Vehicles

Valier Rocket Car

Opel RAK.3 Car Opel RAK.1 Plane

Opel RAK.6 Automobile Opel RAK.1 Airplane https://www.youtube.com/watch?v= https://www.youtube.com/watch?v LmQIWpTW-W8 =vsqg28y_s3s 18

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Russian Aviation Day,

§ K-7 Bomber Prototype 1933-Present § Never flew § Designer executed

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Orbital Motion

Specific Energy… … is the energy per unit of the satellite’s mass § Specific potential energy:

m µ µ P1 PE = − = − S m r r § Specific kinetic energy:

1 m 2 1 2 KES = v = v 2 m 2 P2 § Specific total energy:

µ 1 2 ES = PES + KES = − + v r 2 22

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“Vis Viva (Living Force) Integral”

§ Velocity is a function of radius and specific energy (drop subscript)

1 2 µ ⎛ µ ⎞ v = + E v = 2 + E 2 r ⎝⎜ r ⎠⎟

§ Specific total energy is inversely µ proportional to the semi-major axis E = − (see App. C.4, Sellers) 2a

§ Velocity is a function of radius and semi-major axis (see ⎛ 2 1⎞ http://en.wikipedia.org/wiki/Vis- v = µ⎜ − ⎟ viva_equation) ⎝ r a⎠

Orbital Period

rd • From Kepler’s 3 Law, a3 Period of the Orbit, P, P = 2π , min is (see App. C.6, Sellers) µ • Thus, the orbital −µ2 period is related to P = π , min the specific total 2E 3 energy as where E < 0 for an ellipse

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Examples of Circular Orbit Periods for Earth and Moon

Period, min Altitude above Surface, km Earth Moon 0 84.5 108.5 100 86.5 118 1000 105.1 214.6 10000 347.7 1905

Linear Momentum of a Particle

⎡ ⎤ vx (t) ⎢ ⎥ p(t) = mv(t) = m ⎢ vy (t) ⎥ ⎢ ⎥ v t ⎣⎢ z ( ) ⎦⎥

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Angular Momentum of a Particle (Point Mass)

⎛ dr⎞ h = (r × mv) = m(r × v) = m⎜ r × ⎟ ⎝ dt ⎠ 27

Angular Momentum of a Particle • Moment of linear momentum of a particle – Mass times components of the velocity that are perpendicular to the moment arm h = (r × mv) = m(r × v) • Cross Product: Evaluation of a determinant with unit vectors (i, j, k) along axes, (x, y, z) and (vx, vy, vz) projections on to axes

i j k x y z r × v = = (yvz − zvy )i + (zvx − xvz ) j + (xvy − yvx )k

vx vy vz

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Dimension of energy? Scalar (1 x 1) Dimension of linear momentum? Vector (3 x 1) Dimension of angular momentum? Vector (3 x 1)

Cross Product in Column Notation Cross product identifies perpendicular components of r and v

i j k x y z r × v = = (yvz − zvy )i + (zvx − xvz ) j + (xvy − yvx )k

vx vy vz

⎡ yv − zv ⎤ ⎢ ( z y ) ⎥ Column ⎢ ⎥ r × v = (zvx − xvz ) notation ⎢ ⎥ ⎢ xv − yv ⎥ ⎣⎢ ( y x ) ⎦⎥ 30

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Can We Define a Cross- Product-Equivalent Matrix? Cross product

⎡ yv − zv ⎤ ⎡ ⎤ ⎢ ( z y ) ⎥ ⎡ 0 −z y ⎤ vx ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ v r × v = (zvx − xvz ) = ⎢ z 0 −x ⎥ ⎢ y ⎥ ! r" v ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −y x 0 ⎥ v xv − yv ⎣ ⎦ ⎢ z ⎥ ⎣⎢ ( y x ) ⎦⎥ ⎣ ⎦

Cross-product-equivalent matrix ⎡ 0 −z y ⎤ ⎢ ⎥ r! " ⎢ z 0 −x ⎥ ⎢ −y x 0 ⎥ ⎣ ⎦ 31

Angular Momentum Vector is Perpendicular to Both Moment Arm and Velocity

⎡ yv − zv ⎤ ⎢ ( z y ) ⎥ h = mr × v = m ⎢ zv − xv ⎥ ⎢ ( x z ) ⎥ ⎢ xv − yv ⎥ ⎣⎢ ( y x ) ⎦⎥ ⎡ ⎤ ⎡ 0 −z y ⎤ vx ⎢ ⎥⎢ ⎥ v = m ⎢ z 0 −x ⎥⎢ y ⎥ = mr!v ⎢ ⎥⎢ ⎥ −y x 0 vz ⎣ ⎦⎣⎢ ⎦⎥

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Specific Angular Momentum Vector of a Satellite Momentum per unit of satellite’s mass, referenced to center of attraction m dr hS = r × v = r × v = r × ! r" r#

m dt

Equation of Motion for a Particle in an Inverse-Square-Law Field Recall

2 dv d r µ ⎛ rI (t)⎞ µ a(t) = = 2 = !r! = − 2 ⎜ ⎟ = − 3 r(t) dt dt r (t) ⎝ r(t) ⎠ r (t)

… or µ !r!+ 3 r = 0 r 34

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Artful Manipulation Cross Product of Radius with Equation of Motion ⎡ µ ⎤ r × !r!+ 3 r = r ×[0] = 0, or ⎣⎢ r ⎦⎥ µ r × !r!+ (r × r) = 0 r 3 Cross product of a vector with itself is zero (r × r) = r! r = 0 ⎡ 0 -z y ⎤⎡ x ⎤ ⎢ ⎥⎢ ⎥ ⎢ z 0 -x ⎥⎢ y ⎥ = 0 ⎢ -y x 0 ⎥⎢ ⎥ z ⎣⎢ ⎦⎥⎣ ⎦

35 ∴r × !r! = 0

Specific Angular Momentum is Constant

dr dr Similarly, × = 0 ; therefore,

dt dt

d d (r × r!) = (r! × r!) + (r × !r!) = (r × !r!) = (r × v) dt dt

d (r × !r!) = 0 = (r × v); therefore, dt

(r × v) ! hS = Constant

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Orbital Plane is Fixed in Inertial Space

hS = (r × v)

Angular momentum vector is perpendicular to plane of motion

More Artful Manipulation Cross Product of Equation of Motion with Angular Momentum

⎡ µ ⎤ µ !r!+ r × h = !r!× h + r × h = 0 ⎢ 3 ⎥ S S 3 S ⎣ r ⎦ r Rearranging µ µ !r!× hS = − 3 r × hS = − 3 r × (r × r!) r r

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Eccentricity Vector Apply triple vector product identity (see Supplemental Material) µ µ d ⎛ r⎞ !r!× hS = − 3 r × (r × r!) = − 2 (r!r − r r!) = µ ⎜ ⎟ r r dt ⎝ r ⎠

Integrate over time

⎛ r⎞ ⎛ r ⎞ (!r!× hS )dt = r! × hS = µ d = µ + e ∫ ∫ ⎝⎜ r ⎠⎟ ⎝⎜ r ⎠⎟ where e = Constant of integration e = Eccentricity vector (Constant of integration) 39

Significance of Eccentricity Vector

T ⎛ r ⎞ ⎡ ⎛ r ⎞ ⎤ r! × hS − µ⎜ + e⎟ = 0; therefore, ⎢r! × hS − µ⎜ + e⎟ ⎥ hS = 0 ⎝ r ⎠ ⎣ ⎝ r ⎠ ⎦

T T µr hS T (r! × hS ) hS − − µe hS = 0 r 0 0 ∴−µeT h = 0 S § e is perpendicular to angular momentum, which means it lies in the orbital plane § Its angles provide a reference direction for the perigee § Its magnitude is the orbital eccentricity, e 40

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Classical Orbital Elements Parameters of the Planar Ellipse a : Semi-major axis e : Eccentricity

t p : Time of perigee passage Orientation of the Ellipse Ω :Longitude of the Ascending/Descending Node i : Inclination of the Orbital Plane ω: Argument of Perigee

In-plane Parameters of an Elliptical Orbit Dimensions of the orbit

2 p h Semi-latus rectum = µ = h = Magnitude of angular momentum µ 1 µ E = − + v2 = − = Specific total energy r 2 2a e 1 2 Ep Magnitude of eccentricity vector = + µ = p a = 2 = Semi-major axis 1− e 5 3 2 µE = 3.98 ×10 km s ra = a(1+ e) = Apogee radius RE = 6,378 km

rp = a(1− e) = Perigee radius 42

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In-plane Parameters of an Elliptical Orbit

Position and velocity of the satellite p h2 µ r = = 1+ ecosθ 1+ ecosθ θ = True anomaly µ v = (2a − r) ar Period of the orbit

3 P 2 a = π µ

Orientation of an Elliptical Orbit

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Position and Velocity Following Launch Determine Orbital Elements

Identical major axes = Identical orbital periods 45

Planar Orbit Establishment from Measured Radius, Angular Rate, Velocity, and Time

! 2 Given ro,θo,vo,to ! h = ro θo (in Inertial Frame) 2 p= h µ ! θo = vo cosγ o ro 2 vo µ −1 ! E = − γ o = cos roθo vo ( ) 2 ro (flight path angle) µro a = 2 −1 ⎡1 ⎛ p ⎞ ⎤ 2µ − rovo θo = cos ⎢ −1 ⎥ e ⎜ r ⎟ 2 ⎣ ⎝ o ⎠ ⎦ e = 1− p a FPS-16 Radar

−1 ⎛ a − ro ⎞ ψ = cos : Eccentric Anomaly o ⎝⎜ ae ⎠⎟ a3 t p = (ψ o − esinψ o ) µ

rp = a(1− e) 46

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Effect of Launch Latitude on Orbital Parameters

Typical launch inclinations from Wallops Island (Latitude = 38)

• Launch latitude establishes minimum orbital inclination • Time of launch establishes line of nodes • Argument of perigee established by – Launch trajectory 47 – On-orbit adjustment

Typical Satellite Orbits

GPS Constellation 26,600 km

Sun- Synchronous Orbit

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Geo-Synchronous Ground Track

Geo- Synchronous Ground Track 42, 164 km

Marco Polo- 1 & 2

Orbital Lifetime of a Satellite • Aerodynamic drag causes orbit to decay dV C ρV 2S / 2 = − D ≡ −B* ρV 2S / 2 dt m

B* ! CDS / m, Ballistic factor • Air density decreases exponentially with altitude

−h/hscale ρ = ρSLe

ρSL = air density at sea level

h = atmospheric scale height scale • Drag is highest at perigee – Air drag circularizes the orbit • Large change in apogee • Small change in perigee

• Until orbit is ~circular 50 • Final trajectory is a spiral

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Orbital Lifetime of a Satellite • Aerodynamic drag causes energy loss, reducing semi-major axis, a da = − µaB * ρ e−(a−R) / hscale dt SL • Variatio n of a over time

a e−(a−R) / hs t da = − µB * ρ dt ∫ a SL ∫ a0 0

• Time, tdecay, to reach earths surface (a = R) from starting altitude, h0 = a0 – R hscale h0 / hscale tdecay = (e −1) µRB * ρSL 51

NRL Starshine 1 Orbital Decay (2000)

ISS Altitude = 330-435 km

Altitude, km

Month http://www.azinet.com/starshine/descript.htm 52

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Next Time: Early Space Age …the Heavens and the Earth, Ch 2 to 4;

Launch Dynamics & Staging Understanding Space, Sec 9.1, 14.1 (pp. 535- 542), 14.3

Supplemental Material

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Background Math

Triple Vector Product Identity

a × (b × c) ≡ (a i c)b − (a i b)c = aT c b − aT b c ( ) ( ) Dot Product of Radius and Rate dr r i r! = rT r! = r dt 55