Conic sections

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Conic sections
V
m
r
?
M
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• µ Keplers constant GM (m3 s-2)
• h angular momentum constant
• E total energy constant
• e eccentricity

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Semi major axis a
semi minor axis b
ra
rp
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Solution
1
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• Location of perigee (closest approach) can be
  found from 1 by setting ? to zero

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• Location of apogee (furthest departure) can be
  found from 1by setting ? to p (note that ra only
  has real values for e lt 1, circles and ellipses)

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• Combining perigee and apogee give a useful means
  of finding eccentricity

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Semi major axis, a
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Semi minor axis, b
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family of solutions

• The value of eccentricity determines the category
  of orbit obtained
• e 1gives captive orbits
• e 1creates escape trajectories

e gt 0 lt1 elliptical
e 0 circular
e gt 1 hyperbola
e 1 parabola
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Example 1 circular Earth orbit

• e 0, r 6500 km and 40,000 km

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example 2 Connect with elliptical transfer orbit
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Fuel requirement for manouver

• For LH2/LOX rocket with c 4500m/s

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• This was an example of a Hohmann transfer
• It was believed to have the minimum energy
  requirement for moving between co-planar circular
  orbits, as no fuel is wasted on velocity vector
  rotation
• Disproved by US student who postulated the
  gravity assist manoeuvre for an assignment

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Example 3 hyperbolic capture into circular orbit

• Arrival at mars with a relative approach velocity
  (V8) of 3km/sec, with capture into a circular
  orbit at an altitude of 4000km
• Find ?v required

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V8
?v
V2
Vcirc
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Adjust hyperbolic approach so that its perigee is
at the altitude of the required circular orbit.
The required velocity change will then be along
the line of flight and at maximum velocity,
giving the greatest change in E for a given ?v
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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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Example 4 transfer of a satellite left in LEO
inclined at 30 degrees to the equator to GEO

• This requires the same operations as in example
  2, plus an orbital plane change of 30
• the plane change is best done at a point where
  the orbit crosses the equator, where a simple
  velocity rotation will suffice
• It may be done at LEO, before or after perigee
  burn
• or GEO, before or after apogee kick burn
• or it may be combined with either of the 2 burns
• Choose whichever gives the smallest ?v

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LEO options
10130
7834 m/s
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GEO options
3158
Minimum ?v
1650
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Fuel requirement for manouver

• For LH2/LOX rocket with c 4500m/s