SNAP L2 Orbit Eclipse/Antenna Study

Patrick Jelinsky

01/07/2004

Introduction

The SNAP mission is considering an orbit about the second Lagrange Point
(L2) of the Earth-Sun system. The location of L2 is about 1.5x10^6^ km
from the earth along the earth sun line. A diagram of the L2 orbit is
shown in Figure 1.

![](media/image1.wmf)

Figure 1 Schematic of the L2 orbit

There are many advantages to the L2 orbit. The thermal environment is
very stable. There is little (if any) light from the earth or moon to
scatter into the telescope. There are no radiation belts. The main
radiation environment is the Galactic Cosmic Ray background and the
solar wind.

There are also a few disadvantages to the L2 orbit. The orbit is
unstable with an exponential time constant of about 23 days. This means
that stationkeeping thrusters will be required. It is also far from the
earth so a larger antenna with more power will be needed on the
spacecraft and a large dish on ground to receive the signal.

There are several types of L2 orbits. This memo will present the
analysis of many types of orbits to find the best one for SNAP. The
effects that were considered in the study were; earth and lunar eclipses
of the sun (thermal and power effects), lunar eclipses of the earth
(communications black out), and angular rotation of spacecraft needed to
point a fixed antenna dish to the earth.

A discussion of the math/physics of the L2 orbits is given in the
Background section followed by the analysis. The Background section can
be skipped if the reader is not interested.

# Background

There are 5 equilibrium points in a two mass system rotating with the
masses where the forces are zero. The location of the Lagrange Points is
shown in Figure 2.

![](media/image2.wmf)

Figure 2 Lagrange Point Locations

The distance from the earth to the L2 point can be shown to be:

$$\begin{array}{r}
\rho = \gamma_{l}\text{AU} \\
\gamma_{l} = \left\lbrack \frac{\mu}{3} \right\rbrack^{\frac{1}{3}}\left\lbrack 1 + \frac{1}{3}\left\lbrack \frac{\mu}{3} \right\rbrack^{\frac{1}{3}} - \frac{1}{9}\left\lbrack \frac{\mu}{3} \right\rbrack^{\frac{2}{3}} + \ldots \right\rbrack \\
\mu = M_{2}\left( M_{1} + M_{2} \right)
\end{array}$$

For the earth-sun system μ = 3.0404x10^-6^, γ~l~ = 1.0078x10^-2^ and ρ =
1.5077x10^6^ km. Reference \[1\] gives a nice derivation of these
quantities.

The equations of motion can be simplified if we make a reference system
centered at the L2 point, with all distances scaled to the Earth-L2
distance and all times scaled to the earth's rotation. The coordinate
system is shown in Figure 3.

![](media/image3.wmf){width="4.222916666666666in"
height="2.220138888888889in"}

Figure 3 L2 coordinate system

The equations of motion about the L2 point can be shown to be \[3\]:

$$\begin{array}{r}
\overset{¨}{x} - 2\overset{˙}{y} - \left( 1 + 2c_{2} \right)x = \sum_{n = 2}^{\infty}(n + 1)c_{n + 1}\rho^{n}P_{n}(x\rho) \\
\overset{¨}{y} + 2\overset{˙}{x} + (c_{2} - 1)y = \sum_{n = 3}^{\infty}c_{n}{y\rho}^{n - 2}{\overset{\sim}{P}}_{n}(x\rho) \\
\overset{¨}{z} + c_{2}z = \sum_{n = 3}^{\infty}c_{n}{z\rho}^{n - 2}{\overset{\sim}{P}}_{n}(x\rho) \\
c_{n} = \frac{( - 1)^{n}}{\gamma_{l}^{3}}\left\lbrack \mu + \frac{(1 - \mu)\gamma_{l}^{n + 1}}{\left( 1 + \gamma_{l} \right)^{n + 1}} \right\rbrack \\
{\overset{\sim}{P}}_{n}(x\rho) = \sum_{k = 0}^{\left\lbrack \frac{(n - 2)}{2} \right\rbrack}{(3 + 4k - 2n)P_{n - 2k - 2}}(x\rho)
\end{array}$$

Where P~n~ is the nth Legendre polynomial of the first kind and ρ is the
distance from the L2 point. If only the first order components are kept,
the equations of motion are \[2\]:

$$\begin{array}{r}
\overset{¨}{x} - 2\overset{˙}{y} - \left( 1 + 2c_{2} \right)x = 0 \\
\overset{¨}{y} + 2\overset{˙}{x} + (c_{2} - 1)y = 0 \\
\overset{¨}{z} + c_{2}z = 0
\end{array}$$

For the earth-Sun system c~2~ = 3.94052. The "z" equation is just a
harmonic oscillator. The solutions to the equations have a harmonic
motion in the X-Y plane, and an uncoupled harmonic motion in the "Z"
direction with a different period.

The solution to the equations is \[2\]:

$$\begin{array}{r}
x(t) = A_{1}e^{\lambda t} + A_{2}e^{- \lambda t} + A_{x}\text{cos}(\omega t + \varphi) \\
y(t) = \text{cA}_{1}e^{\lambda t} - \text{cA}_{2}e^{- \lambda t} + \text{kA}_{x}\text{sin}(\omega t + \varphi) \\
z(t) = A_{z}\text{cos}(\nu t + \psi)
\end{array}$$

where A~1~, A~2~, A~x~, and A~z~ are arbitrary constants and c,k,ω, λ,
and ν are given by:

$$\begin{array}{r}
\omega = \sqrt{\frac{2 - c_{2} + \sqrt{9c_{2}^{2} - 8c_{2}}}{2}},\nu = \sqrt{c_{2}},c = \frac{\lambda^{2} - 1 - 2c_{2}}{2\lambda} \\
\lambda = \sqrt{\frac{c_{2} - 2 + \sqrt{9c_{2}^{2} - 8c_{2}}}{2}},k = \frac{- \left( \omega^{2} + 1 + 2c_{2} \right)}{2\omega}
\end{array}$$

For the Earth-Sun system, ω = 2.05701, ν = 1.98507, λ = 2.48432, c =
-0.54526, and k=-3.18723. So the orbital period in the X-Y plane is
177.6 days, and the orbital period in the Z direction is 184.0 days.
This leads to a 3 dimensional Lissajous figure for the orbit. These
orbits will be referred to as Lissajous orbits from now on. If A~1~ or
A~2~ are not zero then the satellites location will grow farther from
the L2 point (unstable). In fact the e-folding time is 365.25\*2π/λ =
23.4 days. Folta \[7\] estimates that a stationkeeping maneuver needs to
be done every three months. He also states that a good rule of thumb is
to allow about 4 m/s of ∆v per year for stationkeeping.

An example Lissajous orbit with A~z~ = 264,000 km (0.175 in L2 units)
and Ax = 82,000 km (0.054 in L2 units) (WMAP orbit) is shown in Figure
4. The dark blue curve is the orbital path, the cyan lines are the
projections of the orbital path in the 3 planes. The projected path in
the X-Y plane is an ellipse. The projection in the X-Z, and Y-Z planes
are Lissajous figures. This plots the orbit for a 2 year period. After
about 14 years the orbit approximately repeats itself. Figure 5 shows a
plot of this orbit in the inertial frame. The distance of L2 from the
earth and the excursions of the orbit from L2 have been expanded by a
factor of 10. The dark blue is the earth. The red is the lissajous
orbit. The cyan and magenta are the orbits projected onto the planes.

![](media/image4.wmf){width="4.752083333333333in" height="3.60625in"}

Figure 4 WMAP Lissajous orbit. Dark blue line is orbital path. Cyan
lines are projections of the orbit in the planes.

![](media/image5.wmf){width="5.625in" height="4.376388888888889in"}

Figure 5 WMAP Lissajous orbit in the inertial frame. The dark blue is
the earth orbit. The red is the lissajous orbit. The cyan is the earth
orbit projected onto the planes, and the magenta is the lissajous
projected on the planes. The distance of L2 to the earth and excursions
from L2 have been expanded by a factor of 10.

Richardson \[4\] presented a solution to the equations of motion brought
out to third order. He found that if appropriate A~x~ and A~z~ are
chosen, a closed orbit can be found. The period in the Z axis equals the
period in the X-Y plane. These are called halo orbits. For the earth-sun
system the relationship between A~x~ and A~z~ is:

$$\begin{array}{r}
A_{x} = 0\text{.}\text{14}\sqrt{1 + 5\text{.}\text{756}A_{z}^{2}} \\
A_{y} = 3\text{.}\text{19}A_{x}
\end{array}$$

The equations of motion for system are:

$$\begin{array}{r}
x\left( \tau_{1} \right) = a_{\text{21}}A_{x}^{2} + a_{\text{22}}A_{z}^{2} - A_{x}\text{cos}\left( \tau_{1} \right) + \left( a_{\text{23}}A_{x}^{2} - a_{\text{24}}A_{z}^{2} \right)\text{cos}\left( 2\tau_{1} \right) + A_{x}\left( a_{\text{31}}A_{x}^{2} - a_{\text{32}}A_{z}^{2} \right)\text{cos}\left( 3\tau_{1} \right) \\
y\left( \tau_{1} \right) = \text{kA}_{x}\text{sin}\left( \tau_{1} \right) + \left( b_{\text{21}}A_{x}^{2} - b_{\text{22}}A_{z}^{2} \right)\text{sin}\left( 2\tau_{1} \right) + A_{x}\left( b_{\text{31}}A_{x}^{2} - b_{\text{32}}A_{z}^{2} \right)\text{sin}\left( 3\tau_{1} \right) \\
z\left( \tau_{1} \right) = ( \pm 1)\left\lbrack A_{z}\text{cos}\left( \tau_{1} \right) + d_{\text{21}}A_{x}A_{z}\left( \text{cos}\left( 2\tau_{1} \right) - 3 \right) + A_{z}\left( d_{\text{32}}A_{x}^{2} - d_{\text{31}}A_{z}^{2} \right)\text{cos}\left( 3\tau_{1} \right) \right\rbrack \\
\tau_{1} = \text{λτ} + \phi \\
\tau = \omega t
\end{array}$$

For the earth-sun system a~21~ = -2.053, a~22~ = -0.252, a~23~ = 0.896,
a~24~ = 0.107, a~31~ = 0.781, a~32~ = 0.0837, b~21~ = 0.491, b~22~ =
-0.0627, b~31~ = 0.855, b~32~ = 0.0204, d~21~ = 0.352, d~32~ = 0.0394,
d~31~ = 0.0188. λ , ω and A~x~ are given above. φ, and A~z~ are the free
parameters.

An example halo orbit with Az = 55,000 km (0.036 in L2 units) (JWST) is
shown in Figure 6 .The dark blue curve is the orbital path, the cyan
lines are the projections of the orbital path in the 3 planes. The
projected path in the X-Y plane is a distorted ellipse. The projection
in the Z-Y is an ellipse, and the projection in the X-Z plane is a
curved line. This plots the orbit for a 2 year period. The orbital
period is about 180 days.

![](media/image6.wmf){width="4.624305555555556in"
height="3.6659722222222224in"}

Figure 6 JWST Halo Orbit. Dark blue is the orbital path. Cyan lines are
projection of the orbit in the 3 planes.

# Eclipse Analysis

The equations from the previous section were used along with code that
predicts the location of the earth and moon to analyze the frequency and
depth of eclipses. Three types of eclipses were analyzed, earth blocking
sun, moon blocking sun and moon blocking earth. The first two types of
eclipses will cause power reduction and thermal changes to the
satellite. The third type (moon blocking earth) will cause a
communication outage.

First halo orbits were analyzed. There is only one free parameter A~z~.
A~z~ was varied and the number of eclipses were recorded. The orbit was
simulated for 20 years so a statistical average of the number of
eclipses could be obtained.

Figures 7-9 show the result of the halo analysis. Figure 7 plots the
number of eclipses in 20 years for various Z amplitudes. If A~z~ is
larger than 14,000 km there are no earth eclipses, and if A~z~ is larger
than 55,000 km there are no lunar eclipses. There are an average of
about 4 communications outages per year roughly independent of the Z
amplitude. Figure 8 shows the maximum eclipse length during the
simulation. The maximum earth eclipse is about 1.2 days, the maximum
lunar eclipse is 1.6 days, and the maximum communication blockage is
\~0.25 days independent of amplitude. Figure 9 shows the minimum
transmission of the eclipse during the 20 years. The earth can block the
sun completely for small A~z~, the moon blocks up to 20% of the sun and
15% of the earth.

Figure 10 is a projection of the 14,000 km A~z~ (red), and 55,000 km
(blue) halo orbits onto the Y-Z plane at the L2 point. In addition, the
maximum eclipse of the sun for all the locations in the Y-Z plane was
calculated for a 20 year period. This image is over-plotted on the
figure. Black is no solar light, red is 50% solar light, and white is
100% solar light (no eclipse). The small dark circle in the middle is
due to earth eclipses and the light orange rectangle is due to the moon.
The red curve just misses all earth eclipses and the blue curve misses
all earth and lunar eclipses. This corroborates the data in Figure 7.

![](media/image7.wmf){width="3.720138888888889in"
height="2.922222222222222in"}

Figure 7 Number of Eclipses in 20 years in a halo orbit. Blue are
earth-sun eclipses, red are moon-sun eclipses, and green are moon-earth
eclipses.

![](media/image8.wmf){width="3.5597222222222222in"
height="2.8743055555555554in"}

Figure 8 Maximum length of the eclipse for the halo orbits. Blue are
earth-sun eclipses, red are moon-sun eclipses, and green are moon-earth
eclipses.

![](media/image9.wmf){width="3.863888888888889in"
height="3.1381944444444443in"}

Figure 9 Minimum transmission of the eclipse for a halo orbit. 1.0 means
no eclipse, and 0.0 means object cannot be seen. Blue are earth-sun
eclipses, red are moon-sun eclipses, and green are moon-earth eclipses.

![](media/image10.png){width="6.016666666666667in"
height="1.4972222222222222in"}

Figure 10 Halo orbits projected on Y-Z plane. Color image is worst
eclipse during a 20 year period at that location in the Y-Z plane. The
dark circle in the middle is due to the earth. The yellow rectangle is
caused by lunar eclipses of the sun. The blue curve has Az = 55,000 km,
the red curve has Az = 14,000 km.

Earth eclipses must be avoided since the battery would probably be
drained during the 1 day length of the eclipse. We could accept lunar
eclipses since they only drop \~15% of the sun. But since the Y
amplitude is not much different between the two halo orbits (14,000 and
55,000 km) lunar eclipses might as well be avoided. Figure 11 plots when
the communication outages occur versus how long the outages are. The
number of outages vary from two to eight per year. Since the outages can
be predicted in advance, observations can be planned during the ¼ day of
outage. Also only 1/7^th^ of the earth is blocked so only about 1 out of
7 of the eclipses will actually block the station on earth.

![](media/image11.png){width="5.965972222222222in"
height="1.9881944444444444in"}

Figure 11 Communication outages for 55,000 km halo orbit.

Next Lissajous orbits were analyzed. There are only three free
parameters A~z~, A~x~ , and φ. The orbital pattern approximately repeats
every 13.9 years. There is no way to avoid earth eclipses totally, but
by selecting the phase angle φ earth eclipses can be avoided for about
6.8 years. This is shown in Figure 12 where 7 years of a WMAP type orbit
is shown projected onto the Y-Z plane along with the eclipse depth
image.

![](media/image12.png){width="5.436805555555556in"
height="3.2708333333333335in"}

Figure 12 Lissajous orbit projected on Y-Z plane. Color image is worst
eclipse during a 20 year period at that location in the Y-Z plane. The
dark circle in the middle is due to the earth. The yellow rectangle is
due to the moon. The curve has A~z~ = 264,000 km and A~x~ = 82,000 km.
Each color is one year of orbit. The order is black, blue, cyan, green,
yellow, read and magenta. There is no earth eclipse until the seventh
year.

The first trial was to fix φ to maximize the time from the start to the
first earth eclipse. The ratio of A~x~ to A~z~ was fixed so that
amplitude in the Y direction was the same as the amplitude in the Z
direction. Then A~z~ was varied and the eclipse effects were studied.
The simulation was run for the 13.9 year repetition period. Figures
13-15 show the result of the halo analysis. Figure 13 plots the number
of eclipses for various Z amplitudes. If A~z~ is larger than \~250,000
km there are about 3 earth eclipses. There are always lunar eclipses.
There is an average of less than 1 lunar eclipse per year for A~z~
larger than \~200,000 km. There are about 3 communications outages per
year for A~z~ greater than \~200,000 km. Figure 14 shows the maximum
eclipse length during the simulation. The maximum earth eclipse is about
20 days, the maximum lunar eclipse is 0.4 days, and the maximum
communication blockage is \~0.25 days independent of amplitude. Figure
15 shows the minimum transmission of the eclipse during the simulation.
The earth can block 85% of the sun for small A~z~, the moon blocks about
12% of the sun and 15% of the earth.

![](media/image13.wmf){width="3.865972222222222in" height="2.875in"}

Figure 13 Number of Eclipses in 13.9 years in a Ay=Az Lissajous orbit.
Blue are earth-sun eclipses, red are moon-sun eclipses, and green are
moon-earth eclipses.

![](media/image14.wmf){width="3.865972222222222in"
height="2.8847222222222224in"}

Figure 14 Maximum length of eclipse for a Ay=Az Lissajous orbit. Blue
are earth-sun eclipses, red are moon-sun eclipses, and green are
moon-earth eclipses.

![](media/image15.wmf){width="3.6944444444444446in"
height="2.8472222222222223in"}

Figure 15 Minimum transmission of the eclipse for a Ay=Az Lissajous
orbit. 1.0 means no eclipse, and 0.0 means object cannot be seen. Blue
are earth-sun eclipses, red are moon-sun eclipses, and green are
moon-earth eclipses.

The next trial was to fix φ to maximize the time from the start to the
first earth eclipse. Then A~x~ was set to 264,000 km (WMAP) and A~z~ was
varied and the eclipse effects were studied. The simulation was run for
the 13.9 year repetition period. Figures 16-18 show the result of this
analysis. Figure 16 plots the number of eclipses for various Z
amplitudes. If A~z~ is larger than \~250,000 km there are about 3 earth
eclipses. There are always lunar eclipses. There are about 3
communications outages per year for A~z~ greater than \~250,000 km.
Figure 17 shows the maximum eclipse length during the simulation. The
maximum earth eclipse is about 3 days, the maximum lunar eclipse is 0.4
days, and the maximum communication blockage is \~0.25 days independent
of amplitude. Figure 18 shows the minimum transmission of the eclipse
during the simulation. The earth can block 90% of the sun for small
A~z~, the moon blocks about 12% of the sun and 15% of the earth.

![](media/image16.wmf){width="3.8243055555555556in"
height="2.622916666666667in"}

Figure 16 Number of Eclipses in 13.9 years in an Ay=264,000 km Lissajous
orbit. Blue are earth-sun eclipses, red are moon-sun eclipses, and green
are moon-earth eclipses.

![](media/image17.wmf){width="3.7909722222222224in"
height="2.5083333333333333in"}

Figure 17 Maximum length of eclipse for an Ay=264,000 km Lissajous
orbit. Blue are earth-sun eclipses, red are moon-sun eclipses, and green
are moon-earth eclipses.

![](media/image18.wmf){width="3.7909722222222224in"
height="2.501388888888889in"}

Figure 18 Minimum transmission of the eclipse for an Ay=264,000 km
Lissajous orbit. 1.0 means no eclipse, and 0.0 means object cannot be
seen. Blue are earth-sun eclipses, red are moon-sun eclipses, and green
are moon-earth eclipses.

The best Lissajous orbit would have Az \> 250,000 km and Ay \> 250,000
km to reduce the number of earth eclipses and maximize the time between
eclipses to about 6 years. There still would be about 1 lunar eclipse
per year. Cobos and Masemount \[2\] calculate the size of maneuvers
required to change the phase of a Lissajous orbits. This can be used to
avoid earth eclipses. One additional maneuver would need to be done
about every 6 years. To minimize the fuel the maneuver should be done at
the corner of the Lissajous just before the eclipse would occur. Two
types of maneuvers are analyzed. The "z maneuver" strategy reverses the
z velocity at the corner of the Lissajous. And the "x-y maneuver" adds a
velocity in the X-Y plane at the corner of the Lissajous. Both maneuvers
reverse the direction of the lissajous, so the path is retraced
backwards to avoid the eclipse. Cobos and Masemount \[2\] show:

$$\begin{array}{r}
\left| \Delta\overset{˙}{z} \right| \simeq \frac{4\text{.}4 \times \text{10}^{6}}{A_{y}(\text{km})}\frac{m}{s} \\
|\Delta v| \simeq \frac{4\text{.}0 \times \text{10}^{6}}{A_{z}(\text{km})}\frac{m}{s}
\end{array}$$

where the first equation is for a "z maneuver" and the second is for a
"x-y maneuver". For Az = 264,000 km and Ay = 264,000 km (WMAP), the
minimum maneuver would be a "x-y maneuver" with a magnitude of 15.2 m/s.

The 264,000 km lissajous orbit has fewer communication outages than the
55,000 km halo orbit. The spacecraft spends more time out of the
lunar-earth eclipse region (see Figure 12). Figure 19 plots the length
of communication outages versus when they occur. The number of outages
vary from one to four per year. Since the outages can be predicted in
advance, observations can be planned during the ¼ day of outage. Also
only 1/7^th^ of the earth is blocked so only about 1 out of 7 of the
eclipses will actually block the station on earth.

![](media/image19.png){width="5.247916666666667in"
height="2.009027777777778in"}

Figure 19 Communication outages for 264,000 km Lissajous orbit.

Antenna Angle Analysis

The antenna for the L2 version of SNAP is fixed. The observatory must be
pointed to direct the dish to earth. Figure 20 shows the location of the
antenna in the SNAP observatory global coordinate system. The system is
defined in reference 5.

![](media/image20.png){width="5.997222222222222in"
height="2.2020833333333334in"}

Figure 20 SNAP coordinate system.

The next analysis was to find the angles needed to point the antenna to
the earth. The default orientation of SNAP is to have the Z axis point
at the ecliptic pole, and the X axis point at the sun. To point the
antenna at the earth from the default position, SNAP must be rolled θ
about the Z axis, and pitched ϕ about the Y axis. Figure 21 shows the
antenna angles required to point to the earth for a 55,000 km halo orbit
(shown in Figure 10). The maximum θ required is \~26°, the maximum ϕ
required is \~2°. Figure 22 shows the antenna angles required to point
to the earth for a 264,000 km lissajous orbit (shown in Figure 12). The
maximum θ required is \~10°, the maximum ϕ required is \~10°. These are
the angles from the default position but SNAP is allowed to point with θ
up to ±45° and ϕ up to ±20°. So the maximum angle change from
observation to communication is ∆θ = 71° and ∆ϕ = 22° for the halo
orbit, and ∆θ = 55° and ∆ϕ = 30° for the lissajous orbit.

There really is no need to decrease the size of the lissajous orbit to
reduce the antenna angles. The angles are already smaller then the SNAP
pointing angles. If the size is decreased more ∆v is required to avoid
eclipses and the time between eclipses will be slightly reduced.

Next the rate of rotation for tracking the earth was checked. SNAP is
specified to have a maximum slew rate or 3°/min \[6\]. For a 1.2 meter
diameter dish using the Ka band (26 GHz), the half power beam width is
about 0.7°. So SNAP needs to track the earth with an accuracy of about
0.2°. The 0.2° should be easy to achieve with the shutter open or closed
since SNAP is required to point to 3" with the shutter closed and 0.8"
(roll) with it open \[6\]. The maximum slew rates for pointing the
antenna were 8x10^-4^ °/min for the halo orbit and 2.7x10^-4^ °/min for
the lissajous orbit. These are well within the SNAP requirements.

![](media/image21.wmf){width="3.9881944444444444in"
height="2.959722222222222in"}

Figure 21 Required Antenna angles for a 55,000 km halo orbit for one
year. blue is for θ, red is for ϕ.

![](media/image22.wmf){width="3.9381944444444446in" height="2.91875in"}

Figure 22 Required Antenna angles for a 264,000 km lissajous orbit for
one year. blue is for θ, red is for ϕ.

# Conclusion

There really is not much difference between the 55,000 km halo orbit and
the 264,000 km lissajous orbit. Each has some advantages and
disadvantages. The advantage of the halo orbit is there are no eclipses.
The advantage of the Lissajous orbit is that it requires 16° less slew
for communications.

The first driver in the L2 orbit decision will be the amount thermal
drift caused by the communications slews. The current plan is to
communicate to the earth once per day for 2 hours. The slew will cause a
different thermal environment on SNAP. The next analysis will be a
thermal analysis of SNAP to see if a slew of 71° for 2 hours every 24
hours will cause thermal distortions of the image. Once this analysis is
done, the type of L2 orbit can be chosen. The largest slew that SNAP
will make actually occurs for observations. It occurs when it changes θ
from +45° to -45°. This will happen once every 90 days and needs to be
done due to the changing solar angle. A thermal analysis of the 90 slew
will also be done to see how long it will take to stabilize after a slew
before observations can begin.

The other driver in the decision is the amount of ∆v required to reach
each type of orbit. An analysis of the launch/maneuver options should be
done to see how much fuel is needed on board SNAP and which vehicles can
be used for the various L2 orbits.

# References

1.  <http://map.gsfc.nasa.gov/ContentMedia/lagrange.pdf>

2.  Cobos, Jordi and Masdemont, Josep, 2002,"Astrodynamical Applications
    of Invariant Manifolds Associated with Collinear Lissajous Libration
    Orbits. <http://www.ieec.fcr.es/libpoint/viewgraphs.html>.

3.  Richardson, David L. *Celestial Mechanics*, **22**, 231, 1980, "A
    Note on a Lagrangian Formulation for Motion about the Collinear
    Points."

4.  Richardson, David L., *Celestial Mechanics*, **22**, 241, 1980,
    "Analytic Construction of Periodic Orbits About the Collinear
    Points."

5.  SNAP drawing 00016-MW02-B,
    <http://sprg.ssl.berkeley.edu/sprite/snap_cm/1_SNAP_Controlled_Drawings_Repository/00016-MW02-B.doc>

6.  SNAP drawing 00007-PP02-A.ppt,
    <http://sprg.ssl.berkeley.edu/sprite/snap_cm/1_SNAP_Controlled_Drawings_Repository/00007-PP02-A.ppt>

7.  Folta, David, "Libration Orbit Seminar",
    <http://gncc.gsfc.nasa.gov/>, or BSCW/PS1000 Mission/L2 Halo Orbit
    Background/David Folta's GSFC Seminar.