L2 Amplitude: Antenna Slew Angle Implications

Patrick Jelinsky

06/28/2004

Introduction

SNAP currently has a fixed high gain antenna. The observatory must be
slewed away from the science target to point the antenna at the earth
whenever communications are to occur. The current plan is to communicate
once per day for about 2 hours.

The amount of onboard fuel required decreases significantly as the
amplitude of the L2 halo orbit increases \[1\]. As the amplitude of the
orbit increases the slew angle to point the antenna at the earth also
increases. Eventually the antenna slew angles become so large that the
sun can shine into the baffle.

This report will summarize the study on how large an amplitude L2 halo
orbit is allowed due to the antenna angle constraints.

# Constraints

There are two pointing constraints that have been used in this analysis.
The SNAP observatory coordinate system \[2\] is shown in Figure 1. The
pointing constraints are shown in Figure 2. The sun vector projected
into the X-Y plane must be within ±45° of the X axis. This is driven by
the "mowing the lawn" supernovae surveys near the ecliptic poles. The
second constraint is that the sun must be within ±25° of the X-Y plane.
This is driven by the location of the surveys near the pole (have to be
accessible all year long), and the angle that the outer baffle \[3\] is
cut.

![](media/image1.png){width="5.997222222222222in"
height="2.2020833333333334in"}

Figure 1 SNAP Observatory coordinate system.

![](media/image2.png){width="4.1125in" height="3.953472222222222in"}

Figure 2 SNAP pointing constraints. The roll must be within ±45° of the
X axis. The pitch must be within ±25° of the X-Y plane.

# Antenna Pointing

Example halo orbits of various amplitudes are shown in Figure 3 \[1\].
The figure shows both the orbit and their projection into the three
planes. The projection into the X-Z plane for the largest orbit is shown
in Figure 4 along with the earth and SNAP pointing the antenna at earth.
The drawing is to scale (except for SNAP). The pitch angle is the same
as that shown in the constraints shown in Figure 2. The projection into
the X-Y plane of the largest orbit is shown in Figure 5 along with the
earth and SNAP pointing the high gain antenna at the earth. Again the
drawing is to scale and the roll angle is the same shown in the
constraints shown in Figure 2.

![](media/image3.wmf){width="4.470138888888889in"
height="4.267361111111111in"}

Figure 3 Halo orbits with different Z amplitudes. Az=522,000 km for
Blue/cyan curve, Az=350,000 km for the red/magenta curve, and Az=55,000
km for the green/yellow curve.

![](media/image4.png){width="4.876388888888889in"
height="3.0458333333333334in"}

Figure 4 Communications pointing projected into X-Z plane.

![](media/image5.png){width="4.345833333333333in"
height="3.6034722222222224in"}

Figure 5 Communications pointing projected into X-Y plane.

The antenna angles were calculated for many amplitude halo orbits. The
angles vary with position on the orbit. The maximum roll and pitch
angles are plotted in Figure 6. For Z amplitudes from 50,000 to 550,000
km the maximum angles do not violate the constraints. The minimum Z
amplitude considered is 55,000 km since this avoids all eclipses \[5\].
The maximum Z amplitude considered is 522,000 km since this requires a
Delta-V of 0 m/s \[1\]. A Z amplitude of 820,000 km will start to
violate the constraints and have the sun shine on the inside of the
outer baffle.

![](media/image6.wmf){width="4.343055555555556in"
height="3.2006944444444443in"}

Figure 6 Maximum antenna pitch and roll angles versus Z amplitude of the
halo orbit.

# Slews

Next the size of the slew from the science target to the earth pointing
was calculated. A large slew can cause two types of problems. First
large slews take longer and therefore take time away from science.
Second since the communications orientation with respect to the sun is
different than the science target, the thermal conditions will be
different. This may cause a thermal drift during the science operations.
Which could degrade the performance.

The slews were calculated using quaternions \[4\]. The location on the
halo orbit was determined every day for six months (one orbit). About
120 possible science target orientations, within 25° of the north
ecliptic pole, were used as the starting point of the slew. The end
point of the slew was the orientation of SNAP that would point the X
axis to the earth but minimize the slew for that starting orientation.
For each day the orientation with the largest slew to earth pointing was
used. Figure 7 shows the results of this analysis. The black curve is if
SNAP just stayed at the L2 point. This is impossible but it shows how
much of the slews are due to science target variations versus moving
along the halo orbit. Even fixed at the L2 location, slews of 50° are
required. The green, red and blue curves are for Az=55,000, 350,000, and
522,000 km respectively.

![](media/image7.wmf){width="5.182638888888889in"
height="4.034027777777778in"}

Figure 7 Maximum antenna slew angles during mission.

The current ACS specification \[7\] has a maximum slew rate of 3° per
second. The smallest halo orbits require a maximum slew of 74° and the
largest halo orbits require a slew of 82°. The smallest halo would
require up to 49 minutes per day to slew to and from the communications
orientation. The largest halo would require 55 minutes per day to slew
to and from the communications orientation. The difference is 6 minutes
per day which is very small compared to the 2 hours per day to download
the data. In reality, SNAP should be able to slew much faster than this.
The moment of inertia of SNAP is about 3500 kg m^2^ in the x and y
directions and 1200 kg m^2^ in the z axis \[6\]. The current momentum
wheels have a limit of 30 N m s. The estimated slew time for 74° is 5
minutes if the wheels are allowed to be ½ full and 10 minutes if the
wheels are allowed to be ¼ full. For an 82° slew the times are 5.6
minutes and 11.2 minutes for ½ and ¼ full wheels respectively. Therefore
the maximum difference between the smallest and largest halo orbits is
2.4 minutes per day.

The science target orientation is different than the communications
orientation. Some areas will have solar illumination in one orientation
and not the other, and some areas will be in darkness in one orientation
and not the other. This may cause different thermal gradients in the
observatory, which could lead to degraded science. The larger the slew
between the orientations, the more likely there will be a problem. The
smallest L2 orbit {fixed at the L2 point) will require a slew of 50° and
the largest halo orbit requires slews up to 82°. It is unlikely that a
thermal design will work for 50° slews and not 82°slews. The more likely
case is that a thermal design will either work for both or not work for
either. If the thermal design works for both then thermal is not an
issue in picking an orbit. If it does not work for either, than SNAP
will require a gimbaled antenna so the orientation of SNAP does not have
to change to communicate.

# Conclusion

The pointing of the observatory for communications will not affect the
choice of the L2 orbit. Halo orbits with Z amplitude up to 522,000 km do
not violate the solar constraints nor do they have much difference in
observation time. The thermal design needs to be refined more to
determine the effect of the communications slews. If the gradients
induced by the slews affect the science, then a gimbaled antenna will be
required.

# References

1)  Jelinsky, P., "Fuel Cost to go to L2", SNAP-TECH-04017, 2004,
    <http://snap.lbl.gov/pub/bscw.cgi/d102307/SNAP-TECH-04017.doc>.

2)  Heetderks, H., "SNAP Observatory Coordinate Systems Definitions",
    00016-MW02.doc, 2003,
    <http://snap.lbl.gov/pub/bscw.cgi/d93154/00016-MW02.doc>.

3)  Sholl, M., "Baffle slice calculations", 2004,
    [http://ssl.berkeley.edu/\~sholl/various/baffle%20slice%20calc%20V0.pdf](http://ssl.berkeley.edu/~sholl/various/baffle slice calc V0.pdf)

4)  Wertz, J. R., "Spacecraft Attitude Determination and Control", 2003.

5)  Jelinsky, P., "SNAP L2 Orbit Eclipse/Antenna Study",
    SNAP-TECH-04010.doc, 2004,
    <http://snap.lbl.gov/bscw/bscw.cgi/d99302/SNAP-TECH-04010.doc>.

6)  Heetderks, H., "SNAP Mass Properties Tracking Spreadsheet",
    00004-ME02-B.xls, 2003,
    http://sprg.ssl.berkeley.edu/sprite/snap_cm/1_SNAP_Controlled_Drawings_Repository/00004-ME02-B.xls