Lecture 2 Conic Sections

8/12/2019 Lecture 2 Conic Sections

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Con ic sect ions

r

M

m

V


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= Keplers constant = GM(m3s-2)

h = angular momentum = = constant

E = total energy = cons tant =

e = eccentr ic i ty =

2r

r

V 2

2

1

12

cos1

2

22

Eh

rh


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ra r

p

semi minor

axis b

Semi major axis a


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Solut ion

21 1 coser h 1


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Locat ion of per igee (closest app roach) can

be found from 1 by set t ing to zero:

2

1

p

hr

e


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Locat ion of apogee (furthest departure) can be

found f rom 1by sett ing to (no te that raonly

has real values fo r e < 1, circ les and el lipses)

2

1a

h

r e


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Combin ing perigee and apogee give a useful

means of f inding eccentr ic i ty :

pa

pa

ap

rrrre

ererh

112


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Semi major axis, a =

2

212

1

p ahr r

e


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Semi minor axis, b =

21p ar r a e


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fam i ly o f solut ions

The value of eccentricity determines the

category of orbit obtained

e 1gives captive orbits

e 1creates escape trajectories

e = 0

circular

e > 0 1 hyperbola


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example 2: Connect w ith

el l ipt ical trans fer orb it


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72.046500

650040000

pa

pa

rr

rre

10

1 cos 1 1

3.88 14*6.5 6*1.72 6.59*10

p ah r e r e r e

E E


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smEr

hV

smEr

h

V

r

hV

p

p

a

a

/130,1065.6

10*59.6

/650,1640

10*59.6

10

10


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sm

sm

sm

v

v

v

/3804

/150816503158

/2296783410130

2

1


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Fuel requ irement for manouver

For LH2/LOX rocket

with c = 4500m/s

429.04500

3804exp

0

m

mf


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This was an example of a Hohmann transfer

It was bel ieved to have the min imum energyrequirement for moving between co -planar

c i rcu lar orb i ts , as no fuel is wasted onveloci ty vector rotation

Disproved by US student who pos tulated thegrav ity ass is t manoeuvre for an ass ignment


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Example 3: hyperbo l ic capture

into circu lar orb i t

Arr ival at mars w ith a relat ive app roach

veloc i ty (V) o f 3km /sec, w ith cap tu re

into a circular orb i t at an al t itude of

4000km

Findvrequired


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V

v

V2

Vcirc


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2

26

2

40008*10 /

2

VE

r

J kg

Adjust hyperbolic approach so that its per igee is at

the alti tude of the required circular orbit. The

requi red velocity change wil l then be along theline of f l ight and at maximum velocity, giving the

greatest change in E for a given v


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smE

EE

rEV

/611064

1327.4*268*2

2

22


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smE

E

rVcirc

/327064

1327.4


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Combinations of these manoeuvres enable all

bodies in the plane of the ecliptic to be visited.

Some moons rotate out of plane, and require

3D calculations of a more complex nature


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LEO opt ions

7834 m/s

10130


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GEO options

1650

3158

Minimum v


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Fuel requ irement for manouver

For LH2/LOX rocket

with c = 4500m/s

28.04500

5724exp

0

m

mf

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Lecture 2 Conic Sections