Scatter Sources for SNAP

Patrick Jelinsky

07/01/2004

Introduction

When designing the baffles for a telescope, the brightness of the
scatter sources must be known to be quantitative. The major sources are
zodiacal light, star/diffuse galaxy light, the moon, the earth, and the
sun. The zodiacal light, star/diffuse galaxy light, and the sun will not
depend on the orbit selection. But the moon and the earth intensity will
depend on the orbit selection. This memo will summarize the intensity of
these scatter sources.

# Zodiacal Light

One on the major scattering sources is the zodiacal light. The zodiacal
light is caused by sunlight scattered by the interplanetary dust
particles. The zodiacal light is a function of the viewing direction
(-sun, ), wavelength, and distance from the sun. It does not vary with
the solar cycle. The zodiacal light depends as R^-2.5^\[1\] where R is
the distance to the sun in AU. Therefore it is not a function of which
L2 orbit is chosen. In fact, since the earth to L2 distance is about
0.01AU the zodiacal light at L2 will be about 97.5% of the zodiacal
light at earth. Figure 1 shows the distribution of zodiacal light from
the data of Leinert et al. \[1\]. The x axis is the ecliptic longitude
minus the ecliptic longitude of the sun. The y axis is the ecliptic
latitude. The intensity is on a log scale. The units of intensity are
the number of 10^th^ magnitude stars per square degree. Reference 1 had
no data within 15 of the sun. An extrapolation of the data was used
there. The intensity varies from 50 to 10^5^ 10^th^ magnitude stars per
square degree.

![](media/image1.wmf){width="5.654861111111111in"
height="2.6069444444444443in"}

Figure 1 Map of the distribution of zodiacal light. The intensity is on
a log scale.

Since the zodiacal light is diffuse, the total of the light that can
enter the outer baffle will be calculated. Over 1,900 orientations of
SNAP that are allowed by the pitch and roll constraints were analyzed.
The constraints that were used were a maximum roll angle 45 and a
maximum pitch angle of 25. See reference 3 for a description of the
angles and constraints. The maximum brightness of the zodiacal light
that enters the outer baffle was equivalent to a -8.1 magnitude star,
the minimum was equivalent to a --5.8 magnitude star, and the average
was equivalent to a --6.1 magnitude star. If the telescope does not
attenuate this light and it is uniformly distributed over the focal
plane, then each 10.5 m pixel will have a count rate of \~2,000
electrons per second. The telescope is likely to attenuate the light by
over a factor of a million.

# Stars

The brightness of the starlight from the sky was estimated using the
Pioneer measurements of the sky \[1\]. This data does not include stars
brighter than 6.5 magnitudes. The Tycho catalog \[2\] was used to add
the stars with magnitudes brighter than 6.5. The resulting map is shown
in Figure 2. The map is dimmer at the poles since the total magnitude in
each bin is displayed and the bin size (square degrees) decreases
towards the poles. The plane of the galaxy is seen along with the large
Magellanic cloud and the constellation Orion.

# ![](media/image2.wmf){width="5.90625in" height="2.6395833333333334in"}

Figure 2 Brightness of sky not including zodiacal light. The intensity
is stellar magnitudes in each bin.

Since the starlight is diffuse, the total of the light that can enter
the outer baffle will be calculated. Over 80 target locations over the
sky were followed for a year. Each day 9 orientations for each target
that was allowed by the pitch and roll constraints were analyzed. The
constraints that were used were same as those used in the zodiacal
analysis. The maximum brightness of the starlight that enters the outer
baffle was equivalent to a -6.4 magnitude star, the minimum was
equivalent to a --6.1 magnitude star, and the average was equivalent to
a --6.3 magnitude star. If the telescope does not attenuate this light
and it is uniformly distributed over the focal plane, then each 10.5 m
pixel will have a count rate of \~2,500 electrons per second. The
telescope is likely to attenuate the light by over a factor of a
million.

# Sun

The brightest scatter source in the sky is the sun. It has a magnitude
of --26.8. The solar light will be limited by cutting the outer baffle
at the proper angle \[4\] so the sun can never shine into it. Also two
baffle edges will be used so double diffraction is needed for the solar
light to enter the outer baffle. The sun is about 9x10^7^ times brighter
than the amount of light entering the baffle from the sky.

# Earth

The earth is second brightest scatter source. The brightness of the
earth is given by the following:

$$M_{\text{earth}} = - 3\text{.}\text{48} + 5\text{log}_{\text{10}}(r) - 2\text{.}5\text{log}_{\text{10}}\left( 1 + \text{cos}\theta \right)$$

where *M~earth~* is the magnitude of the earth, *r* is the distance to
the earth in AU, is the angle between the earth-sun vector and the
earth-satellite vector. = 0 is the full earth, and =180 is a "new"
earth. This was derived from the albedo of the earth and assuming
Lambertian scattering. Figure 3 shows the result of this applied to
calculate the brightness of the earth as a function of orbital phase for
halo orbits with various z amplitudes. The "dips" in the brightness of
the earth versus phase are caused when the earth is very near the sun.

![](media/image3.wmf){width="5.497222222222222in" height="4.14375in"}

Figure 3 Magnitude of Earth as a function of orbital phase and Z
amplitude of the halo orbit.

The maximum brightness of the earth versus z amplitude is shown in
Figure 4. The maximum brightness of the earth varies by less than a
magnitude for halo orbits with a Z amplitude between 50,000 km and
550,000 km. The maximum brightness of the earth is about 80 times
brighter than the amount of light entering the baffle from the sky.

![](media/image4.wmf){width="4.375in" height="3.433333333333333in"}

Figure 4 Maximum Earth Brightness versus Z amplitude of the halo orbit.

# Moon

The brightness of the moon is given by \[5\]:

$$M_{\text{moon}} = 0\text{.}\text{23} + 5\text{log}_{\text{10}}(r) + P(\theta)$$

where *M~moon~* is the magnitude of the moon, *r* is the distance to the
moon in AU, is the moon-sun to moon-satellite angle, and *P()* is the
lunar phase law given by Table 1. = 0 is the full moon, and =180 is a
"new" moon.

Table 1 Lunar Phase Law.

  ------------- ------------------- ----------------- -------------------
                **P()**                               **P()**

  0             0.00                80                2.24

  5             0.08                90                2.63

  10            0.23                100               3.04

  20            0.51                110               3.48

  30            0.79                120               3.93

  40            1.06                130               4.44

  50            1.35                140               5.07

  60            1.62                150               5.90

  70            1.91                160               7.50
  ------------- ------------------- ----------------- -------------------

Figure 5 shows the result of this model applied to calculate the
brightness of the moon as a function of orbital phase for halo orbits
with various z amplitudes. The "dips" in the brightness of the moon
versus phase are caused when the moon is very near the sun.

![](media/image5.wmf){width="5.9944444444444445in"
height="4.936111111111111in"}

Figure 5 Magnitude of Moon as a function of orbital phase and Z
amplitude of the halo orbit.

The maximum brightness of the moon versus z amplitude is shown in Figure
6. The maximum brightness of the moon varies by less than a magnitude
for halo orbits with a Z amplitude between 50,000 km and 550,000 km. The
maximum brightness of the moon is about 1/3 the brightness of the amount
of light entering the baffle from the sky.

![](media/image6.wmf){width="4.471527777777778in"
height="3.801388888888889in"}

Figure 6 Maximum Moon Brightness versus Z amplitude of the halo orbit.

# Conclusion

The scatter source brightness will not be a driver in selecting the
amplitude of the L2 halo orbit. Only the brightness of the earth and
moon depend on the amplitude of the halo orbit. But the brightness
varies by less than a magnitude.

Once the point source transmission (PST) function is determined for SNAP
a more quantitative estimate of the scattered light on the focal plane
can be determined with the data from this memo.

# References

1)  Leinert, Ch, et al., "The 1997 reference of diffuse night sky
    brightness", *A&A*, **127**, 1, 1998.

2)  Hoeg, E., et al., "The TYCHO catalog", *A&A*, **323**, L57, 1997.

3)  Jelinsky, P. "L2 Amplitude: Antenna Slew Angle Implications",
    SNAP-TECH-04018.doc,
    <http://snap.lbl.gov/bscw/bscw.cgi/d102548/SNAP-TECH-04018.doc>.

4)  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)

5)  Larson, W. and Wertz, J., "Space Mission Analysis and Design",
    second edition, 1992, p. 808.