Givens Operational Procedure

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Interplanetary Mission Design ASEN 5519 Spring
2008 Lecture 2 1. Application of Patch Conic to
Deep Impact 2. Orbit Change maneuvers a. Energy
Change b. Inclination Change c. Node and
Inclination Change 3. Sphere Of Influence
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Temple 1 Ephemeris

• ClassificationJupiter-family CometOsculating
  Orbital Elements
• Epoch Julian day 2453440.5, 2005 March 11
  000000.0 UT, Friday
• Semi major axis 3.1219 AU
• Period Julian yrs 5.516 Julian years
• Inclination 10.530
• Eccentricity 0.5176
• Lon. of ascending node 68.941
• Argument of perihelion 178.838
• Passage of perihelion 2005-Jul-05.316301
• Aphelion distance 4.7378 AU
• Mean anomaly 339.217 Physical Data
• Radius 3.10 kmDiscovery Circumstances
• Discoverer Ernst Wilhelm Leberecht Tempel
• Discovered1867-04-03

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Orbit Configuratoin
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Application of Hohmann Transfer to the orbit of
Deep Impact and its encounter of Temple 1
Assume that the Deep Impact spacecraft is on a
Hohmann Transfer.
.
1) Energy of Temple orbit
Launch 1/12/05
2) Compute the velocity of Deep Impact at
perihelion
Impact 7/4/05
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Orbit of Deep Impact spacecraft
Energy
Velocity at aphelion of Deep Impact spacecraft
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Orbit of Deep Impact spacecraft
If we assume all orbits are in the ecliptic the
impact velocity of the spacecraft will be
(neglecting the mass of Temple 1)
The mass of Temple 1 is ,
Hence,
and the due to the mass of Temple 1 is
insignificant.
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Orbit of Deep Impact spacecraft
However, we must account for the inclination of
Temple 1s orbit relative to the ecliptic.
From the law of cosines
Actual value of was
Hence, the Hohmann Transfer is a good
approximation to the actual value of
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Maneuver to change semimajor axis (Energy)
In general, if we are trying to increase or
decrease the energy (semimajor axis) of an
elliptical orbit, the most efficient place to do
this is periapsis.
From the energy equation
Hence, for a given , will be
minimized when V has its maximum value
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Inclination Change
If it is only desired to change the inclination,
the most efficient place to do this is at the
nodal crossing (the only points where the orbits
intersect).
From the law of cosines
If there is to be no Energy change, and
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Inclination Change
From Eq. (2) it is seen that the most efficient
place to change inclination is at the point of
lowest velocity, i.e. apoapsis. Hence the
bielliptic transfer which changes the orbital
energy is also an ideal opportunity for an
inclination plane change. Note that the
are applied at a nodal crossing and transfers are
180 Hohmanns.
Low energy plane change opportunity.
If , then
the bielliptic transfer is more efficient than a
single Hohmann Transfer.
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General Plane Change Both O i
Final Orbit
Initial Orbit
Equatorial Plane
From the law of cosines for spherical triangles
Assuming no energy change the equation for
is given by Eq. 2
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General Plane Change (cont.)
The argument of latitude on the initial orbit at
which the maneuver is performed is given by the
law of cosines for spherical triangles.
Note that is independent of the sign of
since A
reference for Eqns (3 )and (4) is C. D. Brown,
Spacecraft Mission Design, 2nd Edition. Note
that Eq (3.10) of Brown, which is based on the
law of sines, should be replaced by Eq (4) which
yields the proper quadrant.
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Example
Consider an initial circular orbit
It is desired to make a plane change to
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Example (cont.)
The argument of latitude on the original orbit is,
thus
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Example (cont.)
Note that if and
But
Now the argument of latitude is gt 90. Draw a
sketch to show that in this case.
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Example (cont.)
If , then
i.e. The plane change occurs at a nodal crossing.
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Sphere of Influence of the Earth
Ref-Curtis, H.D., Orbital Mechanics for
Engineering Students (p354)
Inertial Frame
We wish to find to describe the motion of the
satellite relative to the Earth.
or
(1)
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Sphere of Influence of the Earth
From Newtons 2nd law
so
(2)
(3)
Subst (2) (3) into (1)
(4)
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Sphere of Influence of the Earth
We may ignore
but
and
Hence,
or
Perturbation acceleration
The ratio of the magnitude of the perturbing
acceleration to the primary acceleration is
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Sphere of Influence of the Earth
Likewise if we consider the spacecraft motion
relative to the sun,
compared to
And the ratio of the magnitudes is,
, using
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Sphere of Influence of the Earth
The sphere of influence (SOI) is the sphere about
the primary body where the perturbations of the
sun on an orbiter of the primary are equal to the
perturbations of the primary on an orbiter of the
sun.
Hence,
setting
, so
But
The sphere of influence for the Earth is,
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Sphere of Influence of the Earth
The SOI, Patch Conic Technique does not work well
for Earth-Moon system because of the perturbing
effects of the sun.