16.07
Lab
II
Issued:
Monday,
November
2,
2009
Due:
Monday,
November
23,
2009
Introduction
In
this
second
numerical
laboratory
exercise,
you
will
build
upon
the
simulation
developed
in
the
first
lab.
Here,
you
will
use
your
capability
to
change
from
the
Earthfixed
inertial
reference
frame,
to
the
EarthMoon
rotating
reference
frames,
in
which
the
Earth
and
Moon
are
fixed
in
the
frame
of
reference.
This
frame
change
will
help
you
explore
different
orbital
phenomena
within
the
framework
of
the
restricted
threebody
problem,
such
as
freereturn
trajectories
to
the
moon,
the
behavior
of
orbits
in
the
EarthMoon
neighbor
as
well
as
the
behavior
of
spacecraft
near
the
Lagrange
points,
both
stable
(L4,
L5)
and
unstable
(L1,
L2).
In
the
restricted
threebody
problem,
we
consider
the
motion
of
three
bodies,
but
do
not
consider
the
gravitational
force
from
the
smaller
body,
the
spacecraft,
on
the
two
large
bodies,
called
the
primaries.
In
lab
1,
you
used
the
radius
of
the
earth
and
your
length
scale,
and
1
day
(24
hours)
as
your
time
scale.
This
results
in
the
following
values
for
the
important
parameters
of
the
system.
Table
1:
NonDimensional
Physical
and
Orbital
Data
µ
earth
11468
µ
moon
141
Rdistance
between
moon
and
earth
60.269
r
0
distance
between
earth
and
origin
.7324
r
1
distance
between
moon
and
origin
59.53
Ωrotation
rate
of
coordinate
system
.230325
If
you
were
successful
in
getting
the
earth
and
the
moon
to
remain
fixed
in
your
rotating
coordinate
system
by
adjusting
slightly
the
value
of
Ω,
you
may
place
the
earth
and
moon
at
these
fixed
points
and
only
consider
the
motion
of
the
spacecraft
in
the
gravitational
fields
of
the
earth
and
moon
in
the
rotating
coordinate
system
using
this
value
of
Ω.
(Since
we
are
going
to
ignore
the
effects
of
the
spacecraft
on
the
earth
and
moon,
we
leave
the
earth
and
moon
at
these
fixed
positions.)
If
you
did
not
obtain
fixed
positions
for
the
earth
and
the
moon
in
the
rotating
coordinate
system,
get
cracking.
I
don’t
think
Lab
2
will
behave
without
this.
1
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Problems
In
the
following
problems,
the
items
you
must
turn
in—either
the
written
answers
to
questions,
or
required
plots—are
highlighted
in
bold.
The
ordering
of
these
deliverables
in
your
report
should
match
the
ordering
that
appears
below.
1.Earth
Fixed
Satellites
In
this
problem,
we
revisit
the
restricted
threebody
problem
simulation
from
Lab
I.
Our
first
task
is
to
place
a
satellite
in
a
circular
orbit
at
a
radius
R
from
the
center
of
the
earth
of
magnitude
R
= 2
R
earth
.
The
proper
choice
of
boundary
conditions
for
this
calculation
is
subtle.
As
shown
in
the
figure,
in
an
inertial
coordinate
system,
placing
a
satellite
in
earth
orbit
requires
matching
the
boundary
condition
with
the
rotation
of
the
earthmoon
system
about
its
center
of
mass.
Consider
for
simplicity
only
the
insertion
point
directly
beyond
the
earth,
on
the
line
joining
the
earthmoon
centers,
the
line
of
symmetry.
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 Dynamics, insertion point

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