Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 3
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 4
Johannes Kepler, born 1571 in Weil der Stadt,mathematician and astronomer
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 5
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!
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 6
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 7
Julius Robert Mayer, born 1814 in Heilbronn,physician (and self-educated physicist)
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 8
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 9
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 10
Walter Hohmann, born 1880 in Hardheim,engineer
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 11
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)
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 12
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)
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 Page 13
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 16
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⋅=
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 17
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!
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 18
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 19
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!
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 20
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 21
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 …
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 22
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 23
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 !
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 24
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 !
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 25
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
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 26
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!
K. Rauschnabel , REVA seminar for CVA teachers, Heilbronn, 6.7.2009 page 27
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]