Orbit calculations using energy and angular momentum conservation

Orbit calculations using energy and angular momentum conservation

Prof. Dr.Kurt Rauschnabel, Mechatronics and Microsystems EngineeringREVA seminar for CVA teachers, 6.7.2009

A few examples related to space flight taken from our physics course for engineering students

K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 2

Welcome to Heilbronnand northern Baden-Württemberg


Heilbronn and the surrounding area –“the cradle of human space flight”

Lampoldshausen (1959 …)

?

?

LampoldshausenHeilbronnJulius Robert Mayer, 1814-1878

HeilbronnWeil der Stadt(≈50 km south of HN)Johannes Kepler, 1571-1630

Weil der Stadt

Hardheim(≈ 50 km north of HN)Walter Hohmann, 1880-1945

Hardheim


Johannes Kepler, born 1571 in Weil der Stadt,mathematician and astronomer


Johannes Kepler, 1571 - 1630

Analysis of astronomical observationsMotion of planets marsComplicated tracks as seen from earthTransformation from geocentric to heliocentric systemFound elliptical orbits (after ≈40 other assumptions)Astronomia novapublished 16092009 – year of astronomy!


Kepler’s laws of planetary motion

1. Orbit of planet is an ellipse,sun at a focus

2. Line joining planet and sun:equal area in equal time

3. Square of orbital periodproportional to the cube of semi-major axis of orbit

.const~ 32

22

31

2132 ===⇒ K

aT

aTaT 2 a


Julius Robert Mayer, born 1814 in Heilbronn,physician (and self-educated physicist)


Julius Robert von Mayer, 1814 - 1878

Heat as a form of energyConservation of energyMechanical heat equivalent(thermodyn. calculation using known heat capacities cp, cv)Published in 1841/1842(1 year before James P. Joule)

later:More precise determination of mechanical heat equivalent

less known:Angular momentum transfer earth-moon by tidal friction


Significance ofJulius Robert von Mayer’s findings

Heat as a form of energyConservation of Energy

Most important fundamental law of modern physicsTechnical application:Conversion of heat to mechanical energy


Walter Hohmann, born 1880 in Hardheim,engineer


Walter Hohmann, 1880 – 1945,engineer and astronautics pioneer

Private work on spaceflight problems (in1911…1915!)Application of well-known laws of physics (Kepler’s laws)1925 publication of his book:Die Erreichbarkeit der Himmelskörper(The accessibility of the Celestial Bodies)

Member of „Verein für Raumschiffahrt“Publications and talks about space transportationCorrespondence with other space and rocket pioneers (Oberth, Ziolkowski, Esnault-Pelterie, Valier)


Walter Hohmann,astronautics pioneer

Escape velocity (needed to leave earth) Detailed orbit calculations for interplanetary flights to mars and venusFuel-efficient path between two different orbits (Hohmann transfer orbit)Re-entry into atmosphereProposal of separate landing vehicle (as used for moon-landing!)Orientation of spaceship by angular momentum conservation (manually - today: use gyroscopes)


Walter Hohmann

HonoursHohmann crater on moon

Wernher von Braunread Hohmann’s book when he was 18recognized (like many other experts) Hohman’s pioneering workHohmann school, Hohmannobervatory, …

accessibilityaccessibility

K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 15

A few examples of orbit calculations

Engineering students, first year Not aerospace engineering, not physics! No frills !

Keep it simple! Do not use space-expert’s terms! Do not use complicated mathematical methods!

Simplifications, e.g.

only 2-body.problems (“heavy” planet plus orbiting satellite) no non-gravitational effects earth as perfect sphere, km 6370=ER gravitational acceleration on surface is g0 = 9.81 m/s² rotation period of earth T = 24 h (exact value: 23 h 56 min 4.1 s) …

Orbit calculation give examples of non-trivial application of fundamental laws of physics


Circular earth orbit Simple example, because radius and velocity are constant energy and angular momentum are constant

Newton’s law of gravity: ( ) 2rmmGrF E⋅

=

need mass of earth: kg 10975 24 .mE ⋅= and grav. constant: 2211 kg/Nm 1067.6 −⋅=G

alternatively: use only radius of earth and grav. acceleration on surface: for ERr = we have ( ) 0mgRF E =

Newton’s law states ( ) 21~r

rF ( )2

0 ⎟⎠⎞

⎜⎝⎛=

rRmgrF E

231420 s/m 1098.3 ⋅==⋅=μ EE RgmG

“standard gravitational parameter”

Using µ, we have: ( ) 21r

mrF ⋅μ⋅=


Circular earth orbit

Approach for orbit calculation: Gravitational force = centripetal force

rmr

m 22

1ω=⋅μ⋅ 2

3 ω=μr

(rv

T=

π=ω

2 is the angular velocity)

222 vrr

=ω=μ

rv μ=

2

32

⎟⎠⎞

⎜⎝⎛ π

=μ

Tr

or .const4 2

3

2

=μπ

=rT

This is Kepler’s 3rd law for circular orbits!


Circular earth orbit

Exercises for students: 1. The ISS is on a circular “low earth orbit” (LEO) at

height km 350=h above ground. Calculate the period T and velocity v !

h 5.123

≈μ

⋅π=rT , km/s 7.7≈

μ=

rv

2. GPS satellites have an orbital period of h12=T . What is the radius of the orbit of a GPS-satellite?

km 266004

32

2

≈πμ

=Tr

3. A TV-satellite is on a geostationary orbit (GSO, period equals earth rotation time) above the equator. Calculate radius and velocity! Can you watch satellite-TV in northern Greenland (latitude λ ≈ 83°) ?

km 422004

32

2

≈πμ

=Tr , °≈λ=λ 81,cos maxmax r

RE 83° is too far north

km/s 1.32≈

π=

Trv

GPS

ISS

GSO


a a

b

b

Elliptical orbits e.g. Hohmann transfer orbit LEO → GSO

Kepler’s first law:

Orbit is a conic section, i.e. circle, ellipse, parabola or hyperbola

depending on starting position / velocity / direction of vr -vector central body is at one focus

No proof given in physics lecture for engineering students!

Newton’s cannonball elliptical orbits are at least plausible!


Angular momentum / angular momentum conservation Angular momentum ( )vmrprL rrrrr

×=×=

Describes rotation of a body. Inertia torque Mr

needed to change Lr

: MtL rr

=dd

Absence of external torque Angular momentum conservation .const=Lr

Kepler’s second law: equivalent to angular momentum conservation!

( ) ( ) ( ) ( )tvtrtvtrtA rrrr ×=α⋅⋅= 2

121 sin

dd

( ) ( ).const

22dd

==×

=mL

mtvmtr

tA

rrr

Note: From this relation we will calculate

the orbital period for an elliptical orbit!

Period:

mL

ba

tA

AT

2dd r

⋅⋅π== (with: area of ellipse baA ⋅⋅π= )

equal area in equal time!

( )trr

α( ) ttv d⋅r

( ) ( ) α⋅⋅⋅= sindd 21 ttvtrA rr


a

bA B

C

D

F 1rr

ab

2rr

1vr

2vr

3vr3rr

Angular momentum / angular momentum conservation Lr

can easily be calculated at points A, B, C, D:

in A ( )11 vr rr ⊥ : 11 mvrL ⋅=r

in B ( )22 vr rr ⊥ : 22 mvrL ⋅=r

in C (D): 3mvbL ⋅=r

compare A (perigee), B (apogee,):

2211 vmrvmrL ⋅=⋅=r

or 1

2

2

1

rr

vv =

e.g. for a Hohmann transfer orbit LEO → GSO we get

km 67201 =r , km 422002 =r 3.62

1 ≈vv

satellite looses a factor ≈6 in velocity when “climbing” from A to B ! gains a factor ≈6 in velocity when “falling” from B to A!

To calculate 21, vv we need energy conservation …


a a

a b

b

Elliptical earth orbit Energy conservation: .const=+= kinpottot EEE

Potential energy is not “mgh ” !

We have to integrate Newton’s law of gravity ( ) 21r

mrF ⋅μ⋅= ( EmG ⋅=μ )

choosing ( ) 0=∞potE we get:

( )rmr

rmrE

r

potμ−

=μ= ∫∞

d12

Energy conserv. at A → B: 2

222

1

1

212

1

rmmv

rmmv μ−=μ−

Relation 21 vv ↔ (angular mom. conserv. : 2

112 r

rvv = ): 2

22

212

121

1

212

1

rm

rrmv

rmmv μ−=μ−

Calculate kinetic energy and total energy, e.g. at r1

⎟⎟⎠

⎞⎜⎜⎝

⎛−μ=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

212

2

212

121 111

rrm

rrmv ( )

22

21

22

21212

11

11

rrrrrmmvrEkin −

−μ==

A B


Kinetic energy at r1: ( )2

2

21

22

21212

11

11

rrrrrmmvrEkin −

−μ==

Total energy at r1: ( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

μ=μ

−=1

21

22

22

21

1

212

11

111

rrr

rrr

mr

mmvrEtot

( ) ( )

( )( ) ( )( )( ) 12121

122

1121

2

1121221

2212

1

1

11

rrm

rrrrrrm

rrrrrm

rrrrrrrrrrmrEtot

+−

μ=++−

μ=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+μ=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

+−−

μ=

Using arr 212 =+ (major axis of ellipse!) we get the very important result: amEtot 2

μ−=

Total energy on elliptical orbit does only depend on semi major axis a !


ab

C

Fa

b

3vra

We can now easily calculate speed as a function of r:

am

rmmvEtot 2

221 μ−

=μ

−= ⎟⎠⎞

⎜⎝⎛ −μ=

arv

21122

“vis-viva equation”

Let us now go back and calculate the angular momentum (at point C), now using the vis-viva equation

Note: aFCr ==3

a

mbaa

mbmvbL μ⋅=⎟

⎠⎞

⎜⎝⎛ −μ⋅=⋅=

21123

r

Finally we calculate the orbiting period for the elliptical orbit:

amb

mbaT

mL

ba

tA

ATμ⋅

⋅⋅⋅π=

⋅⋅π==

2,

2dd r Important: T only dep. on semi major axis!

Thus we have derived Kepler’s 3rd law: .const4 2

3

2

=μπ

=aT

for elliptical orbits !


Exercise for students: 4. Hohmann transfer LEO → GSO (cf. examples1 and 3)

given: km 3501 =h km672011 =+= hRr E , km/s 7.7=LEOv km 422002 =r , km/s 1.3=GSOv Calculate:

a and b of Hohmann ellipse, Time needed for transfer (=T/2), Velocity for elliptical orbit at 21, rr , vΔ for LEO→ transf. orbit and transf. orbit → GSO

ERrra ⋅≈=+

= 8.3km 244702

21 ,

km 177501 ≈−== raeFM , ER.eab ⋅≈≈−= 62km 1684422

h 6.1023

≈μ

π=aT , h 3.52

1 ≈T

at 1r : km/s 1.102112

11 ≈⎟⎟

⎠

⎞⎜⎜⎝

⎛−μ=

arv , km/s 4.21 +≈Δv

at 2r : km/s 6,12

112 ≈=

rrvv , km/s5.12 +≈Δv

a a

a b

b F M


Exercise for students: 5. Toolbox-orbit

On November 18, 2008 space shuttle Endeavour astronaut Heidemarie Stefanyshyn-Piper lost a toolbox during work outside of the ISS. assume: ISS on LEO , km3501 =h , km672011 =+= hRr E , km/s7.7=LEOv toolbox escaped backwards with m/s5.1−=Δv calculate:

min. and max. height of toolbox-orbit, diff. in orbital period (toolbox – ISS)

vis-viva eq. ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛μΔ+−

=

212

12

1

vvr

aLEO

, km6.21 −=−=Δ raa , km8.344212 =Δ+= ahh

orbit “350 km x 344.8 km” as 2

3

~ aT we can use error propagation” to calculate TΔ :

s 2.323

−=Δ⇒Δ⋅=

Δ Taa

TT (toolbox orbits faster!)

TΔ is equivalent to a distance of km25≈ after one orbit!


Thank you for your attention! Are there any questions ?

Please use our potential well (and your €-coins) for orbit simulations

Content of potential well goes to”KRAKI”, the nursery of

Prof. Dr. Kurt Rauschnabel Hochschule Heilbronn Mechatronik und Mikrosystemtechnik [email protected]

Documents

Orbit calculations using energy and angular momentum conservation