Introduction to Orbital Mechanics: AV, the Conic Sections, and Kepler's

Home , Hyperbolic trajectory

Introduction to Orbital Mechanics: !V, the Conic Sections, and Kepler’s First Law “Its not where you are, its how fast you are going”

Isp=400sec Isp=300sec !V Isp=200sec

m0/mf Sutton and Biblarz, Chapter 4

MAE 5540 - Propulsion Systems Example 1: Isaac Newton explains how to launch a Satellite

MAE 5540 - Propulsion Systems Example 2: Sub Orbital Launch

• Still “in orbit” Around earth center

• What is velocity At apogee is Zero?

• Do we still need !V?

MAE 5540 - Propulsion Systems How Much Delta-V do you need?

!Vrequired = !VburnoutOrbit + !Vgravity + !Vdrag Thrust, Weight, Lift, Drag

Yes!

MAE 5540 - Propulsion Systems Rocket Design 101

…. How Much !V do you need to accomplish the mission?

!Vrequired = !VburnoutOrbit + !Vgravity + !Vdrag

• Obviously we have a LOT! To learn about !V!

MAE 5540 - Propulsion Systems Gravitational Physics

Gravitational Physics • Now a bit of "gravitational physics"

Isaac Newton, (1642-1727)

MAE 5540 - Propulsion Systems Orbital Velocity • Object in orbit is actually inOrbi ta"free-fall"l Velocity that is ... the object is literally falling around the Earth (or Planet) • When the Centrifugal Force of the "free-fall" counters the Gravitational Force ... the object Fcentrifugal is said to have achieved Orbital Velocity

Fgrav

MAE 5540 - Propulsion Systems N-body problem

MAE 5540 - Propulsion Systems Gravitational Attraction on a 10,000 kg Spacecraft 100

Geo 10 Attraction of Earth

0.1 Attraction of Sun

0.01 Attractive Force, Newtons

0.001 Attraction of Moon 10000 20000 30000 40000 Orbital Altitude, kilometers

MAE 5540 - Propulsion Systems MAE 5540 - Propulsion Systems MAE 5540 - Propulsion Systems Two Body Problem

MAE 5540 - Propulsion Systems MAE 5540 - Propulsion Systems Kepler’s laws

MAE 5540 - Propulsion Systems Conic Sections

MAE 5540 - Propulsion Systems Circular Orbits: Circular orbits

MAE 5540 - Propulsion Systems i

MAE 5540 - Propulsion Systems Circular Orbits

MAE 5540 - Propulsion Systems Elliptical Orbits: Elliptical orbits

• Adding "!V" turns a circular orbit into an elliptical orbit

!VT

MAE 5540 - Propulsion Systems What is an Ellipse? y semi-minor axis

apfocus b x y ra rp ! a x (-c,o) (c,o) c "line of apsides" semi-major axis perifocus

Geometry of an Ellipse

ra + rp = 2 a

MAE 5540 - Propulsion Systems Ellipse Equation: Earth Centered at origin y

b rp ra y ! a x (-2c,0) (-c,0) (0,0) c

MAE 5540 - Propulsion Systems Polar-Form of the Ellipse Equation

j x+c x+2c X

! r a y r p

a i (-2c,o) (-c,o) (o,o) c c Geometry of an Ellipse (Earth Centered at perifocus)

MAE 5540 - Propulsion Systems Polar-Form of the Ellipse Equation (cont'd)

MAE 5540 - Propulsion Systems Polar-Form of Ellipse Equation (cont'd)

• Substituting and simplifyiing

MAE 5540 - Propulsion Systems Polar-FormPolar Form of Ellipse Equation of the Ellipse Equation (concluded) • Defining the elliptical eccentricity as

• The polar form of the ellipse equation reduces to

MAE 5540 - Propulsion Systems Parameters of the Elliptical Orbit

MAE 5540 - Propulsion Systems Fundamental Elliptical Orbit Deﬁnitions

apogee Ra Rp perigee 2b “Line of Apsides”

2a a -- Semi-major axis b -- Semi-minor axis e -- Orbital eccentricity

MAE 5540 - Propulsion Systems Fundamental Elliptical Orbit Deﬁnitions cont’d

Orbit Apogee and Perigee (closest and farthest approaches) … semi major axis and eccentricity related to apogee and perigee radius

MAE 5540 - Propulsion Systems Fundamental Elliptical Orbit Deﬁnitions cont’d

apogee Ra Rp perigee 2b

2a Location within an Orbit: “true anomaly” "

MAE 5540 - Propulsion Systems CoordinateCoordinate Transformations:Transformations j ! i r {i, j} ﬁxed in space i! ! i

MAE 5540 - Propulsion Systems Coordinate Transformations:(cont;d) Coordinate Transformations • A Matrix "trick" for coordinate transform in 2-D j ! i r i! ! i

MAE 5540 - Propulsion Systems Orbital Velocity

Velocity is the derivative of position ….

2 a(1! e ) r {a,e} % constant rr = " i $ 1+ e " cos(#) r parameters of the orbit

2 2 r d r d & a(1! e ) ) r a(1! e ) d r V = (r ) = ( + " ir + " (ir ) dt dt ' 1+ e " cos(#)* 1+ e " cos(#) dt

MAE 5540 - Propulsion Systems Orbital Velocity (2) Differentiate r d $ a(1! e2 ) ' $ a(1! e2 ) ' & ) = !& 2 ) "(!e "sin(#)"(#&)) = dt % 1+ e " cos(#)( % +*1+ e " cos(#)-, ( 2 $ a(1! e ) ' $ e "sin(#) ' $ e "sin(#) ' & ) "& ) "(#&) = & ) "(r "#&) % +*1+ e " cos(#)-,( % +*1+ e " cos(#)-,( % +*1+ e " cos(#)-,(

r Differentiate i d r d r r r (ir ) = ( i ! cos(") + j !sin(") ) = ( -i !sin(") + j ! cos(") )!"& = "& ! i" dt dt

MAE 5540 - Propulsion Systems Orbital Velocity (3)

2 r ' e !sin(") * r ' a(1- e ) * r V = ) !(r !"&), ! ir + ) , !"& ! i" = ( $#1+ e ! cos(")&% + ( 1+ e ! cos(")+ #' e !sin(") * r r % (r !"&).) , ! ir + i" / $.( $#1+ e ! cos(")&%+ &/

Need Kepler’s Second Law to get

!& = " MAE 5540 - Propulsion Systems MAE 5540 - Propulsion Systems ParabolicParabolic Trajecto riesTrajectories:

MAE 5540 - Propulsion Systems What is a Parabola?

Directrix • Cartesian form of the equation r

y r ! focus (0,0) x p p

• A parabola is the locus of all points equidistant from a fixed point (focus) and a line (Directrix)

MAE 5540 - Propulsion Systems Parabola Equation Polar Form

Use quadratic formula to solve

MAE 5540 - Propulsion Systems Parabola Equation Polar Form (cont'd)

MAE 5540 - Propulsion Systems Parabola Equation

Polar Form (concluded) • r must be > 0, therefore pick - sign

Perigee!

MAE 5540 - Propulsion Systems Hyperbolic HypTrajectories:erbolic trajectories "Excess Hyperbolic Velocity" approach/departure" V! > 0

Parabolic approach/departure" V! = 0

• If "V !" > 0 , then probe will approach and depart along a hyperbolic trajectory

MAE 5540 - Propulsion Systems What is a Hyperbola?

Cartesian Equation !

left focus c b (apfocus) c (2c,0) (0,0) right focus (c,0) (perifocus)

c 2= a2 + b2

• A Hyperbola is the locus of all points whose difference from two fixed points is constant

MAE 5540 - Propulsion Systems Hyperbolic Equation: polar form

a c r ! c b c (2c,0) (0,0) right focus (c,0) (perifocus)

c 2= a2 + b2

MAE 5540 - Propulsion Systems Hyperbolic Equation: polar form (cont'd)

MAE 5540 - Propulsion Systems MAE 5540 - Propulsion Systems Hyperbolic Asymptotes • What is the behavior of a Hyperbola at "infinity" i.e. along away from earth

MAE 5540 - Propulsion Systems MAE 5540 - Propulsion Systems Kepler’s First Law (Summary)

MAE 5540 - Propulsion Systems Homework:2: Enterprise Compound Circular Orbits: i s j i s s !s moon !s as as moon j j p s Tralfamador E nterprise a p ap Tralfamador

!p i p planet Strangelove

MAE 5540 - Propulsion Systems Homework:Compound Orbits (cont'd) • Starship Enterprise orbits alien moon Tralfamador in a circular orbit of radius as

• Moon orbits alien planet Strangelove in circular orbit with radius ap

• Alien GPS system orbiting moon gives position relative to Tralfamadorian -fixed coordinate system. • Due to gravitational damping Tralfamador, always keeps the same face directed towards Strangelove

MAE 5540 - Propulsion Systems Homework:Compound Orbits (cont'd) • Compute the position vector of the Enterprise relative to Strangelove ... in the Strangeloveian -fixed coordinate system -- Rsp

• Solution should have as, ap, !s, !p as parameters

MAE 5540 - Propulsion Systems CoordinateCoordinate Transformations:Transformations j ! i r {i, j} ﬁxed in space i! ! i

MAE 5540 - Propulsion Systems Homework:Circular Orbits (concluded)

• Compute the velocity vector of the Enterprise relative to Strangelove ... in the Strangeloveian -fixed coordinate system.

MAE 5540 - Propulsion Systems