Infrared Limb Sounding With Cassini CIRS

Infrared Limb Sounding With Cassini CIRS:
Optimal Viewing Strategy Using Horizon Nodes
Conor A. Nixon, Richard K. Achterberg,
Department of Astronomy
University of Maryland
College Park, MD 20742
301-286-6757
301-286-1550
[email protected]
[email protected]
F. Michael Flasar
Solar System Exploration Branch
NASA Goddard Space Flight Center
Greenbelt, MD 20771
301-286-3071
[email protected]
Abstract—In this paper1 2 we investigate a question of
science optimization during Cassini flybys of Titan. The
Composite Infrared Spectrometer (CIRS) makes limb
observations – along an atmospheric path above surface –
during the closest approach period when the visible horizon
circle is moving swiftly across the planet. We have sought
to discover if any points on the horizon are preferred for
limb sounding due to having minimum movement relative
to the surface. By numerical calculation, backed by
geometric analysis, we find that two limited regions on the
horizon are continuously visible during the entire encounter.
We term these ‘limb nodes’ and show how they may be
employed by CIRS to optimize science by minimizing the
source of systematic error due to spatial smear. These
conclusions are applicable to many similar scenarios of
spacecraft limb sounding during hyperbolic flyby
encounters.
a substantial atmosphere. During these close approaches,
Cassini’s Composite Infrared Spectrometer (CIRS)
instrument is used to sense the atmosphere in limb-sounding
mode – viewing along a path that does not does intersect the
surface – to measure the vertical variation of temperature,
aerosols (hazes) and gas composition. The opacity of a limb
path, defined as:
TABLE OF CONTENTS
When viewing the limb, the available choice of latitude and
longitude locations is defined by the instantaneous horizon
circle. At large distances (D) an entire hemisphere is visible,
and the horizon is a great circle (circumference) of the
globe. However as the spacecraft approaches to distances
where the range is no longer much greater than the planetary
radius (R), the horizon circle rapidly shrinks and expands
around closest approach, and its center (the sub-spacecraft
track) also moves across the surface. Obtaining each limb
sounding profile near closest approach – when vertical
resolution becomes better than an atmospheric scale height
(the vertical distance in which density drops by 1/e) –
requires a time that is not insignificant compared to the
movement of the projected horizon circle on Titan’s surface.
Therefore, a question of optimization arises: is there a
preferred point (or points) on the horizon to view, where the
lateral movement of the tangent point is least, so that the
vertical profile obtained most closely corresponds to a
single latitude and longitude?
χν ≡
1. INTRODUCTION
Since July 2004, the Cassini spacecraft has been orbiting
Saturn, making more than 60 close flybys of the large
satellite Titan, the only moon in the solar system to possess
2
ν
(1)
[where kν is the spectrally-dependent absorption co-efficient
(cm2/g), ρ is the atmospheric density (g/cm3), and dl is the
path element (cm)] is usually sharply peaked at the tangent
point, €
so long as the emissions are non-saturated (optically
thin, χ<1) and therefore the retrieved values at various
altitudes correspond to the atmospheric column above a
specific latitude and longitude co-ordinate on the surface.
1. INTRODUCTION .................................................................1
2. LIMB VIEWING WITH CASSINI CIRS ..............................2
3. LIMB VIEWING HORIZONS AND NODES ...........................4
4. ANALYTICAL TREATMENT ...............................................7
5. IMPLICATIONS FOR OPERATIONS ..................................11
6. SUMMARY AND CONCLUSIONS .......................................12
ACKNOWLEDGEMENTS ......................................................12
REFERENCES .......................................................................13
BIOGRAPHY ........................................................................13
1
∫ k ρ dl
978-1-4244-3888-4/10/$25.00 ©2010 IEEE.
IEEEAC paper #1174, Version 5, Updated January 4, 2010
1
In this paper we investigate the problem both numerically
and theoretically, showing that optimum solutions exist. Our
recommendations, described in the context of CIRS
sounding of Titan are expected to be of broad applicability
to other remote sensing applications.
the detector of the Voyager IRIS [3]. For middle infrared
(mid-IR) science, a conventional Michelson (amplitude
splitting) interferometer is used, with detection capability
provided by two 1x10 HgCdTe arrays. One array (FP3) is
based on the photo-conductive (PC) principle and is
sensitive from 600-1100 cm-1 (17-9 µm) while the second
(FP4) is operated photo-voltaically and covers the shortest
wavelength region: 1100-1400 cm-1 (9-7 µm).
2. LIMB VIEWING WITH CASSINI CIRS
Cassini Mission and Flybys
Fig. 1 shows the projection of the three focal planes on the
sky with correct relative positions and sizes. Note that the
far-IR bolometer has a much larger field of view (FOV)
than the mid-IR pixels. Its spatial response function
approximates a Gaussian with 50% of integrated response
contained in a diameter of 2.54 mrad, whereas the FP3 and
FP4 detectors have approximately square spatial responses
of FWHM 0.29 mrad. The spatial responses of the FP4 array
are much closer to idealized boxcar functions than those of
FP3, which resemble U-shapes due to details of the solidstate design [4].
As Cassini orbits Saturn, it makes frequent encounters with
the giant moon Titan, serving the dual purposes of enabling
Titan science investigations – including remote sensing, in
situ sensing of fields and particles, radar sensing, gravity
measurements and radio occultations, and more - but also
as a driver for the overall Saturn tour, using Titan’s gravity
to deflect the spacecraft’s path to rendezvous with smaller
satellites and to change the inclination to the ring plane.
Table 1 – Cassini Mission Phases
Mission
Phase
Start Time
Prime
Mission
1-JUL-2004
Equinox
Mission
Solstice
Mission
End Time
Saturn
Orbits
Titan
Flybys
30-JUN2008
75
45
1-JUL-2008
30-SEP2010
63
26
1-OCT2010
16-SEP2017
155
56
The mission has three major phases: the prime mission
(PM), referring to the first four years of operation as
originally planned; the extended or equinox mission (XM or
EM) covering the next 27 months and including the
northern spring equinox of Saturn and Titan in August
2009; and the second extended or solstice mission (XXM or
SM) stretching a further seven years (as currently
envisioned at time of writing) beyond the end of the EM,
and including the northern summer solstice of the Saturnian
system in May 2017. Further details of the Cassini mission
phases can be found in Table 1.
Figure 1 – footprints of CIRS instrument detectors
projected on the sky.
The spectral resolution of CIRS is variable, and because it is
a Fourier Transform interferometer (FTIR), the highest
resolution (smallest spectral interval) is determined by the
maximum path difference of the scan mechanism. Therefore
the spectral resolution can be varied from 15 cm-1 (lowest)
to 0.5 cm-1 (highest), with corresponding scan times of 5 s to
52 s respectively. Note that these spectral resolutions refer
to the full-width to half maximum (FWHM) of the
instrumental line shape (ILS) using Hamming apodization.
Unapodized widths of the primitive sinc line shape are about
half as wide.
CIRS Instrument
The Cassini Composite Infrared Spectrometer (CIRS) is a
hybrid instrument, with two separate interferometers sharing
a common telescope, fore-optics, scan mechanism, reference
laser, and back-end electronics. See [1] for design details.
The far-infrared (far-IR) spectral regime (10-600 cm-1,
1000-17 µm) is a Martin-Puplett [2] type interferometer that
uses wire-grid polarizers to split the incoming radiation. The
recombined signal is then detected by a single large
bolometer detector known as focal plane 1 (FP1), similar to
Nadir versus limb sounding
In normal viewing, the instrument FOV is pointed towards a
planetary surface, however in limb viewing mode, the FOV
is instead directed along an atmospheric path that does not
intercept the surface (or equivalently defined spheroid,
2
normally the 1-bar pressure level, in the case of gas planets
such as Jupiter and Saturn). Fig. 2 shows a schematic of
nadir (A, B) and limb sounding (C) geometries.
determined by the signal-to-noise level at a given
wavenumber: when the emission of a given gas species
drops below the noise level of the instrument then no
information can be retrieved [5]. Limb sounding also
provides a great advantage for the detection and
measurement of very faint trace atmospheric species,
including isotopologues, by increasing the atmospheric path
length substantially beyond what is possible in a ‘nadir’
mode, and thereby accumulating signal for optically thin
emission lines. The principal disadvantage of limb sounding
is the added complexity in modeling the vertically-varying
detector field.
In normal, or ‘nadir’ viewing mode3 (A) the infrared path at
most ‘thermal’ wavelengths (where reflected sunlight is
negligible, about λ>5 µm) becomes optically thick (χ≥1, see
Eq. 1) above the surface. This defines the atmospheric level
at which information will be retrieved from the spectrum.
Increasing the emission angle by looking away from the
sub-spacecraft point (B) increases the atmospheric path, and
raises the altitude of the peak information region slightly.
However, looking off-center towards the disk edge also
carries disadvantages, such as increasing the inhomogeneity
of the field in terms of the latitudes, longitudes and emission
angles included in a detector FOV, making modeling more
difficult.
CIRS limb sounding objectives
For Cassini CIRS, the two main targets for limb viewing are
Saturn and Titan. For each target, the limb viewing is
performed in several varieties, e.g. ‘integrating’ (stationary
dwell at a particular altitude) versus ‘scanning’ (moving the
FOV radially on the limb to sense multiple altitudes).
Depending on the distance to the target and the primary goal
of the particular observation type (e.g. temperature sounding
versus trace gas composition) the spectral resolution (scan
length) and length of dwell or scan angular speed are
adjusted to achieve the scientific result. The various types of
limb sounding performed by CIRS are described in detail in
[6], which also depicts the ‘contribution functions’ showing
the peak altitudes at which information is obtained. In this
paper, we focus on limb sounding of Titan, which is more
complex than Saturn sounding for Cassini as the spacecraft
approaches much closely to Titan, resulting in rapidly
changing geometry during flybys.
Before leaving our discussion of the CIRS limb science
objectives, we must make one further observation regarding
vertical spatial resolution. Distance from Titan (D)
determines the range to the limb (r), according to the
Pythagorean formula: r2=D2+R2, where R is the moon’s
radius of 2575 km. The vertical spatial resolution on the
limb is a function of the detector angular resolution and the
range: Δz=rΔθ, so that we arrive at a formula giving the
distance from Titan required for a given vertical resolution
on the limb:
Figure 2 – schematic comparison of nadir and limb
sounding geometries.
Limb viewing has the principal advantage of allowing
remote sensing over a much wider range of altitudes than is
possible in surface-intercepting mode. The lowest level that
can be sensed is equivalent to the highest altitude probed in
nadir mode (θe=89°). However, many higher altitudes can
be probed by pointing the FOVs at these levels where most
infrared paths remain optically thin (χ≤1). Due to the
exponential decrease in atmospheric density with altitude,
which occurs both before and after the tangent point, the
information is usually sharply peaked at the tangent altitude
itself, achieving the desired result, although the Q-branches
of some strong gas bands (e.g. ν4 band of CH4) may not
probe as deeply as the bands of other less abundant
molecules. The highest altitude that can be probed is
 Δz 2 
D =  2  − R2
 Δθ 
(2)
(replace ‘R’ with ‘R+z’ for rays above the surface). For
CIRS FP1 Δθ=2.54 mrad, whereas for FP3 and FP4
Δθ=0.29 mrad. For vertical profile measurements we often
€ a spatial resolution (Δz) of an atmospheric scale
require
height or better, defined as the vertical distance in which
density drops by a factor 1/e, and equal to about 50 km in
the lower stratosphere. Therefore we compute that a
maximum range of 19,000 km is required for useful far-IR
limb sounding, whereas for mid-IR limb sounding the range
need only be 166,000 km. As the spacecraft encounters
Titan at a relative velocity of typically 20,000 km h-1, this
3
Strictly the term ‘nadir’ should only be applied when the emission angle
is θe=0°, and the line of sight is normal to the surface, but in practice is
applied to all rays that intercept the surface, with θe<90°.
3
shows that the period inside ±60 mins from closest approach
is the most useful for far-IR limb sounding, while for midIR limb sounding we can achieve adequate resolution over a
much wider time span of at least ±8 hrs. This vast difference
between the mid and far-IR requirements is a driver of the
critical need to perform limb sounding, especially inside the
±1 hr period around closest approach for far-IR science,
which also happens to be when the geometry is changing
rapidly. This is the topic of the next section.
3. LIMB VIEWING HORIZONS AND NODES
Horizons
At very large distances from a target body, the view
encompasses an entire hemisphere, but as the range
decreases so too does the amount of visible surface. The
reader may wish to mentally envisage the amount of the
Earth’s surface visible from a low flying aircraft, and
contrast to that seen by a geostationary satellite. We define
the horizon circle or ellipse by the points where emitted rays
reaching the observer are tangential to the surface, with an
emission angle of 90°. In the limit of infinite distance, the
horizon circle delineating the bounds of vision is a great
circle of circumference, but as the observer approaches the
body the horizon circle also shrinks. Fig. 3 shows this
effect. The observer (spacecraft) at position 1 sees a much
larger horizon than at 2.
For a spherical body (such as Titan), the horizon is circular
and centered on the instantaneous sub-spacecraft point: the
point where the imaginary line joining the spacecraft to the
body center intersects the body surface. As the spacecraft
approaches and recedes from the body the sub-spacecraft
point traces an arc on the surface, which is also therefore the
locus of the instantaneous horizon circles of varying sizes.
Figure 3 – the changing size of the visible horizon circles
at different distances from the target body.
The SPICE toolkit routines SPKEZ and EDLIMB were used
in conjunction with the Cassini SP (SPacecraft trajectory), I
(Instrument boresight) and C (spaCecraft attitudes) kernels
to compute the Cassini horizon circles on Titan as a function
of time. Time intervals of ±60, ±30, ±15 and 0 mins relative
to ‘closest approach’ (minimum distance or t=0 for the
flyby) were used for the period where the geometry was
changing most rapidly. The horizon circles were then
plotted as loci of latitude and longitude using an Aitoff
equal-area map projection for every Titan flyby, using the
IDL software package.
Knowledge of the horizon circle is clearly important for
limb sounding: to a first approximation this technique
samples the atmosphere directly above the horizon circles.4
We now show how these can be numerically calculated, and
show that an interesting property arises. In section 4 we will
give a more rigorous mathematical treatment of horizon
circles.
Numerical computation of spacecraft horizons
The time-dependent spacecraft trajectory and pointing
attitude are tabulated in numerical files, provided by the
Cassini navigation team (NAV), known as the SPICE
kernels. SPICE is a standard multi-mission information
system, defined by the JPL Navigation and Ancillary
Information Facility (NAIF), comprising binary data files
for spacecraft ephemeris and pointing data and solar system
ephemerides, and a set of software tools for interrogating
the files to calculate distances, angles etc between various
bodies (spacecraft, planets, moons) [8].
Two examples are shown here. In Fig. 4 we see the horizon
circles for the T15 flyby. Inbound horizons are red, have
centroids near the 0°W longitude line and are wrapped from
the left edge of the image back to the right edge; the horizon
at closest approach is black, the smallest horizon; and the
outbound horizons are blue, increasing in size with time
from left to right. T15 was an equatorial flyby (spacecraft
trajectory along Titan’s equatorial plane) therefore the
horizon circles are also symmetric about the equator.
4
For optically thick spectral regions the emission may saturate before the
tangent point: see [7] for details.
4
In Tables 2 and 3 we give the approximate coordinates of all
the limb nodes for the three phases of the Cassini mission,
computed here as the intersection of the inbound and
outbound horizon circles at t=±30 mins. The latitudes as a
function of time are plotted in Figs. 6 & 7.
Figure 4 – computed horizon circles for the Cassini T15
Titan flyby, at 0, ±15, ±30 and ±60 mins from closest
approach.
Note again that the range of latitudes at which limb
sounding can occur is defined by the latitudes crossed by the
horizon circles, so that the latitude range diminishes towards
closest approach as the horizon shrinks, and the range
increases again after closest approach. We also note an
interesting property of the horizon circles, namely that all
seven circles appear to cross at two coordinates or ‘nodes’:
approximately (50°N, 110°W) and (50°S, 110°W).
In Fig. 5 we see the horizon circles for T26, which was an
‘inclined’ flyby, in contrast the ‘equatorial’ T15. The
spacecraft trajectory was south-to-north, with an approach at
(45°S, 45°W) (red circles) and a departure around (210°W,
45°N). The interesting property of the two horizon circle
crossing points or ‘nodes’ that we noted previously for T15
is also evident here: the nodes are now at (40°N, 45°W) and
(10°N, 280°W).
Figure 5 - computed horizon circles for the Cassini T26
Titan flyby, at 0, ±15, ±30 and ±60 mins from closest
approach.
5
moving surface target. As this air column can vary zonally
(i.e. with longitude, as described in [9]), a systematic error
is introduced.
Figure 6 – computed latitudes of both 30-minute limb
nodes for each Titan flyby in the Cassini prime (through
T44) and equinox missions (T45-T70).
Figure 8 – CIRS far-infrared limb integration
observation for Cassini Titan 15 flyby. The large FP1
bolometer is centered successively at 125 km, 225 km
above the horizon node.
Now the value of the limb nodes is apparent: only these two
specific latitude and longitude locations remain visible on
all horizon circles for the duration of the encounter.
Targeting either of these nodes for limb sounding
guarantees that the same air column will be sampled for the
entire observation, removing a source of systematic error.
Figure 7 – computed latitudes of both 30-minute limb
nodes for each Titan flyby in the Cassini solstice
missions (T71-T126).
Use of horizon nodes: latitude and longitude smear
In fact these horizon nodes have much more than mere
curiosity value, and provide a crucial tool for optimizing the
value of limb sounding information during flybys. Consider
the options available to a CIRS science planner wishing to
perform limb sounding during the closest approach period
on the T15 flyby, at ±1 hrs from t=0. A typical observation
(either scan, or integration-dwell) requires at least 30
minutes or greater to perform, but during that time the
horizon circle also moves. On T15, the spacecraft is moving
close to Titan’s equatorial plane. If the instrument is pointed
for example at 0°N during the outbound period on the righthand limb, it is apparent that the longitude of the horizon
will move from 225°W at +15 mins to 270°W at +60 mins.
Therefore, the instrument is not sampling the same air
column during this time, but the changing column over a
Figure 9 - CIRS far-infrared limb aerosol scan
observation for Cassini Titan 26 flyby. The large FP1
bolometer is moved radially outwards at relative
altitudes from -100 km to 700 km at the horizon nodal
point.
Figs. 8 and 9 show examples of the final designs (from the
Cassini Pointing Design Tool (PDT) software package) for
two types of limb observation: a far-IR integration on
outbound T15, targeted at 50°N on the right-hand limb
6
(relative to north), and a far-IR radial scan at 10°N on the
left-hand limb on outbound T26.
x = a; ( y 2 + z 2 ) = ( R 2 − a 2 )
(4)
where a is the x-coordinate of the plane containing C. By
substitution of y=Rcosθcosφ and z=Rsinθ from (3) into
The surprising constancy of the horizon intersection points € equation (4) we arrive at the spherical coordinate equation
over time prompted us to analyze the problem
for the circle:
geometrically. Our goals were (1) to see if we could predict
the stationary node locations, and (2) to see if any deeper
(5)
r = R; R 2 cos 2 θ cos2 φ = a 2
conclusions could be reached through an exact treatment.
We were successful in both these regards, as now described.
We now consider the position of the observer to find the
value of a.
Horizon Circle in Spherical Coordinates
4. ANALYTICAL TREATMENT
€
We here consider a 2-D spacecraft trajectory past a spherical
target body, where the plane of the spacecraft trajectory also
intersects the body center and bisects the sphere. This is
defined to be the x-y plane, with perpendicular axis z and
origin at the center of the target. The corresponding
spherical polar coordinates of any point p in this system are
(r,θ, φ). Note that for equatorial flybys x-y is the equatorial
plane, and (θ, φ) corresponds to planetocentric latitude and
longitude respectively, but in general this is not the case.
See Fig. 10.
Figure 11 – 2D section of any arbitrary plane
perpendicular to the horizon circle, showing intercepts
with sphere.
Figure 10 – horizon circle on sphere, 3D view.
Fig. 11 shows a cut through the plane containing the
spacecraft (SC), the planet center, and any two opposing
horizon points P and P′ (vertical axis not necessarily y or z).
The spacecraft is located at variable distance D along the +x
axis, the tangent lines to the planetary surface T locate the
horizon points, which subtend angles ψ at the planet center.
We note that:
A spacecraft at SC viewing along the –x axis direction sees
a horizon defined by the circle C, which is co-planar to y-z
plane. P is any point on the horizon circle, and has coordinates (R, θ, φ) in spherical polar co-ordinates, where R
is constant and equal to the planetary radius, θ is the angle
between R and the x-y plane, and φ is the azimuth angle in
the x-y plane measured from the x-axis. Any point on the
surface of the sphere is located by the Cartesian vector:
cosψ =
R = Rcosθ cos φ i + Rcosθ sin φ j + Rsin θ k
(3)
= xi + y j+ zk
and therefore we can eliminate a from Eq. (5) to find:
while C is specified in Cartesian coordinates by:
€
a R
R2
= ⇒a=
R D
D
7
€
(6)
R 2 = D2 cos2 θ cos 2 φ
symmetric (ignore gravity assists etc) about this point and
lies in the x-y plane.
(7)
We must now extend our treatment to the case where the
observer is at any arbitrary position in the x-y plane outside
2
2
2
of
€ the planet (x +y >R , or r>R). Due to rotational
symmetry, we can assert that Eq. (6) holds for any circle
perpendicular to x-y, centered at azimuthal angle φ=α, if we
replace φ in Eq. (7) with Δφ≡φ-α:
R 2 = D2 cos2 θ cos 2 (φ − α )
(8)
Locating the Nodal Region
From
€ Figs. 4 and 5 we have seen that the successive horizon
circles during the inbound and outbound portions of a
typical flyby tend to intersect at two approximate
coordinates, that we have termed ‘nodes’. In reality, these
apparent confluences of horizon circles are rather ‘fuzzy
nodes’ that have some spatial variation except for a unique
case. We have explained the utility of these nodes for limb
viewing previously, so it is our objective here to predict
them spatially.
Figure 12 – front view of two horizons on sphere viewed
along -x-axis.
For a spacecraft flyby, the values of (α, D) are the
coordinates of the time-dependent spacecraft trajectory in
the x-y plane. Therefore, the coordinates of the circle (θ,φ)
also vary with time, and so for a given spacecraft trajectory
we can re-write Eq. 8 as a new function: either θ = f(φ,t) or
φ=(θ,t). The instantaneous nodal points are located where
the circles at t and t+dt (a small time later) intersect. Over
the course of the encounter, these two loci trace a pair of
arcs to either side of the sub-spacecraft track on the surface.
These arcs, limited by some time period ±τ, define the true
boundaries of the fuzzy node during the encounter, but this
problem is trajectory-dependent and not easily solvable (e.g.
the first and subsequent time derivatives of the horizon
circle equation are not necessarily zero) except for a few
special cases.
Fig. 12 shows the view of two horizon circles CI (inbound at
time -t relative to closest approach at t=0, azimuth of center
φc=-α) and CO (outbound at time +t, φc=+α), which are
symmetric in time and space about the x-z plane (x is out of
the page at the origin). These circles necessarily intersect at
points N1 and N2 which are located at spherical coordinates
(R,+θn, 0) and (R,-θn, 0). There is no need to simultaneously
solve the equations of the two circles, as we may simply
note that the nodal points are located where the circles
intersect the x-z plane and φ=0 and hence for CI we have:
R 2 = D2 cos2 θ cos 2 (α )
(9)
with solutions:
However, a much simpler purely geometric approach offers
an alternative route to finding the approximate location of
the nodal region. We previously noted that the nodes are
‘regions of intersection’ of all inbound and outbound
horizon circles. We can therefore seek the coordinates of
intersection of inbound and outbound horizon at symmetric
times and positions about the point of closest approach,
forming a time series, which is a much simpler problem
than a calculus approach. I.e. we will find the intersection of
C(-t) and C(t), rather than C(t) and C(t+dt), for -τ<t<+τ.
Note that we have approximated the true problem here, and
our solution does not permit azimuthal variation of the
nodes.
€
R

θ n = ± cos−1 sec(α ).
D

(10)
Therefore we have found the locations of the two horizon
intersections (R, ±θn, 0) as a function of one-sided timedependent spacecraft position (D, 0, α), where D=D(t) and
€
α=α(t). Note again that the azimuthal variation of the true
time-dependent nodes is not captured, nevertheless we will
show that this approximation is very useful near closest
approach. By inserting an equation for the spacecraft
trajectory, we can then reduce the solution to one (time
dependent) variable.
Linear Trajectory
We define the x-axis as being in the direction of the radius
vector at closest approach (periapse) during the pass, and
make the assumption that the spacecraft trajectory is
We will first consider the simplest possible case, a linear
spacecraft trajectory, which ignores any gravitational effect
of the central body on the spacecraft and is the limiting case
8
for flybys at great distance, high encounter velocity or very
low mass of the target.
Figure 14 – spacecraft trajectory for true hyperbolic
flyby.
Fig. 14 (adapted from Fig. 11.5 of [10]) shows the orbital
geometry. The planet is at the focus F, and the spacecraft at
a radial distance D. As before we denote the periapse
distance as D0, and the angle between D and D0 is α, known
as the true anomaly. The velocity vector in general has
radial (vr) and transverse (vt) components in the polar
coordinate system centered on F. The trajectory is
symmetric about periapse, and as t→±∞ the hyperbolic path
tends to the two asymptotes, which have an opening angle δ.
We will quote here without proof the standard results for the
hyperbolic orbit, showing the relation between α and D that
enables θn to be calculated (Eq. 10). For full derivation see
[10].
Figure 13 – spacecraft geometry for idealized (linear)
flyby.
Fig. 13 shows the generalized location of the spacecraft in
the x-y plane during a linear flyby, and we immediately find
that:
cosα = D0 D
(11)
and so substituting in (10) we have :
€
cosθ n = R D0 .
(12)
This important relation means that in the case of a linear
flyby, the ‘nodal line’ of (10) collapses to two, timeinvariant nodes at (R, ±cos-1(R/D0), 0). Given the fact that
€
many high altitude (D0 ≥ several R) Cassini flybys of Titan
do approximate quite closely to a linear flyby, the apparent
exactness of the nodes (e.g. Fig. 4) is finally explained.
We first define the two constants of motion, the specific
energy E (energy per unit mass, sum of kinetic and
gravitational energy divided by mass) and angular
momentum h:
E=
Hyperbolic trajectory
In general, the spacecraft trajectory past a gravitating body
in an unbound flyby is a hyperbola (ignoring any thruster
firings, rotation of the object that adds or subtracts delta-v,
inhomogeneities in the body’s gravitational field and
relativistic corrections). The proof of this from the inversesquare law of gravitation can be found in standard texts on
orbital mechanics: see Chap. 11 §11.4 of [10].
vp µ
− ; h = Dv t ⇒ h = D0v p
2 D0
(13)
where µ≡GM, the gravitational parameter. Note that E is
positive for unbound orbits, and that the periapse velocity vp
€is purely transverse. Then the parameters of conic section,
the eccentricity e and the semi-major axis a (e>1 and a<0
for all hyperbolae) can be written in terms of the physical
parameters:
9
a=−
µ
D
2ED0
; e = 1− 0 ; ⇒ e = 1+
.
2E
a
µ
If we consider Eq. 16, we note that the cosθn is proportional
to 1/D, and also to 1/D02. As for angles in the range 0-90°,
the cosine decreases as the angle increases, we infer that at
small ranges (D) the separation of the nodal points (2θn) is
also relatively small, compared to large ranges when the
separation becomes larger. Also, the same relationship holds
for the periapse distance and the nodal point angular
separation, except that the effect is much greater due to the
D 02.
(14)
The geometric equation of orbit relating D to α is given by:
€
D=
€

h2 
1


µ 1+ ecos α 
2 2
1  D0 v p 
⇒ cosα = 
−1.
e  Dµ

(15a)
Some examples are illustrative here. The flyby T67 from the
Equinox mission phase is exactly in the equatorial plane of
Titan, but at a relatively high altitude of 7461 km. The
horizons are shown in Fig. 15, and due to the high value of
D0 we predict high values of θn which translate here directly
into a high-latitude nodal point. The T61 flyby was centered
at a low southern latitude (~20°S), but a low altitude of 970
km. Therefore we not only predict that the nodal points will
be much closer together, but also that they will show much
greater change during the ±1 hr period than those on T67.
(15b)
So finally we can write (10) as
€
 µRe 
θ n = ± cos−1 2 2

 D 0 v p - Dµ 
(16)
giving a relation of the nodal position (intersection of
inbound and outbound symmetric horizon circles) as a
function
€ of the spacecraft position on the hyperbolic path.
Unfortunately, the time dependence of the spacecraft
position on a hyperbolic orbit (either D or α) is not given by
a simple analytical function and must be found numerically
(see Eq.s 11.46 through 11.48 of [10]).
Implications and example applications
At the beginning of this section we implied that by solving
(or approximating) the problem analytically we might hope
to gain some useful additional insights into the location of
the limb nodes during flybys, that can be valuable for
science planning and other purposes. We will now show that
this is indeed the case.
Figure 16 - computed horizon circles for the Cassini T61
Titan flyby, at 0, ±15, ±30 and ±60 mins from closest
approach.
The calculated results are shown in Tables 4 and 5. Note
that for this purpose we used tables of spacecraft (D, v) as
functions of time provided by the Cassini project using the
aforementioned NAIF tools, rather than numerically solving
the (D, t) relation. We then computed the eccentricity and
substituted for (e, D(t), D0, vp) in Eq. 16 to obtain θn(t).
Figure 15 - computed horizon circles for the Cassini T67
Titan flyby, at 0, ±15, ±30 and ±60 mins from closest
approach.
Table 4 shows the results for the T67 flyby. The values of
θn(t) change little over the ±1 hr period, and all the horizon
circles intersect very close to 75°N, as seen on Fig. 15. Note
10
Sun.5 This is forbidden during CIRS observations, as the +X
direction hosts a passive radiator that keeps the mid-IR focal
plane assembly at an operating temperature of 80 K.
Therefore, only one attitude of +Z or –Z will be accessible,
and the secondary axis is completely determined for each of
the two limb positions. The final decision as to which of the
two limb positions is observed is determined by
consideration of two additional factors:
that the prediction of the linear flyby approximation is
θn=R/D0 which is identical to θn(0) and equals 75.1°. We
can now easily understand why the nodes on this flyby are
approximately constant, as for the linear model. The reason
is because the high periapse altitude D0 compared to the
change in D over this period leads to little change in θn.
Alternatively we can say that the high eccentricity (35)
causes this flyby to approximate a linear trajectory.
1. Turning time from the ‘waypoint’ secondary axis
orientation (the existing secondary axis when
CIRS picks up control of the spacecraft) to each of
the two limb attitudes.
2. Scientific considerations as to which of the two
locations is preferable. E.g. usually prefer to
observe a latitude that is significantly different to
recent flybys.
Table 5 shows the calculated θn(t) for the T61 flyby, which
was much lower than T67. In this case, θn is changing
significantly over the ±1 hr period, as we see on Fig. 17.
The much lower D0 (or alternatively speaking the lower
eccentricity of 13) means that this flyby is not well
approximated by a simple linear trajectory model (θn(t)
=θn(0)=43.4°, as we expect.
I.e. if the turning times to both limb positions are similar,
then the scientific consideration wins. In a very small
number of cases CIRS has chosen to observe a particular
limb nodal point even though placing either +Z or –Z
perpendicular to the disk are both ruled out by thermal
flight rules and long turns. In that case, the FP3 and FP4
arrays are not in a radial direction, and some mid-IR
science is lost. However this may occasionally be chosen
for reason of either (i) very high priority for far-IR science
at a particular latitude, or (ii) no possibility of placing +Z or
–Z perpendicular to disk at either nodal point.
In summary, the simple analytical model for computing the
time variation of the horizon nodes has given us a deeper
understanding of the locations and relative variations of
these points during the period near closest approach. We can
therefore carry the qualitative knowledge that: (i) higher
flybys will have nodal points further from the sub-spacecraft
line that vary little in time, and (ii) lower flybys will have
nodal points closer to the sub-spacecraft line that vary more
rapidly in time, as we begin to plan the limb operations for
CIRS in a given Titan flyby.
Working with other Cassini instrument teams
CIRS far-IR limb observations occur in the period from 135
minutes to 15 minutes from closest approach, either
approaching or departing the planet. During the prime
mission,6 CIRS preferred to use the time as follows:
5. IMPLICATIONS FOR OPERATIONS
I.
Secondary axis restrictions
CIRS has four defined boresights, one centered on each of
the three focal planes (FP1 bolometer, and the centroids of
FP3 and FP4 arrays), and a further boresight defined as the
combined centroid of FP3 and FP4, designated ‘FPB’. All
four of these axes are near to the –Y axis of the spacecraft.
The arrays of FP3 and FP4 are oriented parallel to the
spacecraft Z-axis as depicted in Fig. 1. For far-IR limb
viewing, the primary axis is always FP1 towards the target
point, but as the detector is circular, FP1 has no preference
for secondary axis orientation. Normally, it is preferable to
choose this secondary axis so that +Z or –Z is perpendicular
to the disk edge, so that the mid-IR arrays are pointed
radially from the disk center. This is so as to obtain
maximum vertical information from FP3 and FP4, which
are observing simultaneously to FP1.
Far-IR limb integration: (135 to 75 minutes,
45,000 to 25,000 km). Two dwells on limb with
FP1 at 125 km, 225 km. Two repetitions at each
position for 12 minutes.
II. Far-IR aerosol scan: (75 to 45 minutes, 25,000 to
15,000 km). Single long scan radial to limb, -100
to +600 km.
III. Far-IR temperature scans: (45 to 15 minutes,
15,000 to 5,000 km). Two shorter radial scans from
0 to 300 km.
In practice, these exact boundaries were frequently adjusted
to fit around other demands on the spacecraft time, such as
RADAR, ultraviolet (UVIS) and visible-near-IR (VIMS)
stellar and solar occultations, imaging (ISS), radio
5
Unless the Sun is exactly behind the spacecraft, at a phase of 0 degrees to
the limb position.
6
Several modified and hybrid observation types are being used in the
equinox and solstice phases of the mission.
In practice, for each of the two limb locations, one or other
direction of +Z or –Z perpendicular to disk will almost
always place the +X direction in the same hemisphere as the
11
occultations and other activities. It was extremely rare for
any of the optical remote sensing teams to command the
entire period ±2 hrs around closest approach – this happened
only once in the mission for CIRS.
•
Therefore, working in this period requires extremely close
cooperation between the various Cassini instrument teams.
A major aspect of this is modeling and timing the turns
between the various requested spacecraft attitudes. Turns
between a CIRS limb attitude and other instruments in the
ORS group (ISS, VIMS, UVIS) that nominally have
boresights close to the –Y axis are usually moderate if (i)
the secondary axes are similar, and (ii) the distance to the
planet is not less than 10,000 km, where the apparent
diameter becomes large and turns between limb and disk
center, or limb to different limb (for occultations) become
non-negligible. Turns between the various ORS instrument,
including CIRS, and other teams such as RADAR are
usually onerous due to the 90° change in primary axis that is
often required.
•
ACKNOWLEDGEMENTS
The authors express their appreciation to the many members
of the Cassini spacecraft team at JPL who work daily to
plan and implement the science and operate the spacecraft.
Also to the members of the CIRS team at GSFC, JPL and
elsewhere who operate the instrument, calibrate,
disseminate and archive the data. Special thanks is due to
Todd Ansty for implementing many of the CIRS Titan limb
observations. The authors acknowledge the support of the
NASA Cassini project during the period in which this work
was performed.
Modeling these turns during the early integration phase
(when the time line is being constructed) can lead to
important pit-falls being discovered, e.g. when a team is
awarded an amount of time that is insufficient to turn to and
from their pointing attitude and perform any meaningful
science. Mitigation is often possible when discovered early
enough, during integration, for example by reallocating
time, but is much more difficult later on during
implementation when the teams are designing their
observations and boundaries are very difficult to shift
without major repercussions.
6. SUMMARY AND CONCLUSIONS
In this paper we have described the nature of limb sounding
with Cassini CIRS near closest approach during Titan
flybys; explaining the particular geometry involved, and
how this gives rise to horizon circles that move over time.
We have sought to answer the question as to whether there
are any particular advantages of observing a specific point
over any other: the answer is an emphatic yes.
By plotting a time series of horizon circles for the
encounter, we showed that two (latitude, longitude) regions
are visible for the entire duration on all horizon circles: we
refer to these as ‘nodes’. By geometric analysis we showed
how these nodes arise, and that in the special case of a linear
trajectory these are exactly two (lat,lon) coordinates. In the
more general case of a hyperbolic flyby the nodes form
small regions. Observing either of these nodes is preferable
over any other horizon point because the same localized
parcel of air will be observed over time, reducing systematic
errors in the retrieval of atmospheric quantities.
We have also noted operational considerations of CIRS farIR limb sounding, including:
•
perpendicular to disk edge, so as to maximize
atmospheric (vertical) structure information from
the FP3 and FP4 mid-IR arrays.
Thermal constraints that can further restrict the
choice of secondary axis.
The need to model and test ‘hand offs’ (turns)
between instruments sharing the ±2 hr closest
approach part of the time line.
The preference for a secondary axis of ±Z
12
REFERENCES
[8] Charles H. Acton Jr., “Ancillary Data Services of NASA’s
Navigation and Ancillary Information Facility”, Planetary
and Space Science, vol. 44, pp. 65-70.1996.
[1] Virgil G. Kunde, P. Ade, R. Barney, D. Bergman, J. F.
Bonnal, R. Borelli, D. Boyd, J. Brasunas, G. Brown, S.
Calcutt, F. Carroll, R. Courtin, J. Cretolle, J. Crooke, M.
Davis, S. Edberg, R. Fettig, F. M. Flasar, D. Glenar, S.
Graham, J. Hagopian, C. Hakun, P. Hayes, L. Herath, L.
Horn, D. Jennings, G. Karpati, C. Kellebenz, B. Lakew, J.
Lindsay, J. Lohr, J. Lyons, R. Martineau, A. Martino, M.
Matsumura, J. McCloskey, T. Melak, G. Michel, A.
Morell, C. Mosier, L. Pack, M. Plants, D. Robinson, L.
Rodriguez, P. Romani, W. Schaefer, S. Schmidt, C.
Trujillo, T. Vellacott, K. Wagner, D. Yun., “Cassini
Infrared Fourier Spectroscopic Investigation,” SPIE 2803,
162-177, 1996.
[9] Richard K. Achterberg, B.J. Conrath, P.J. Gierasch, F.M.
Flasar, C.A. Nixon, “Oberservation of a tilt of Titan's
middle-atmospheric super-rotation,” Icarus 197, 549-555,
2008.
[10] Bruno Bertotti, Paolo Farinalla and David Vokrouhlicky,
The Physics of the Solar System (Second Edition),
Amsterdam: Springer, 2003.
[2] D. H. Martin and E. Puplett, “Polarized interferometric
spectroscopy for the millimeter and submillimeter
spectrum,” Infrared Physics 10, 105-109, 1969.
BIOGRAPHY
[3] R. Hanel, B. Conrath, D. Gautier, P. Gierasch, S. Kumar,
V. Kunde, P. Lowman, W. Maguire, J. Pearl, J. Pirraglia,
C. Ponnamperuma and R. Samuelson, “The Voyager
Infrared Spectroscopy and Radiometry Investigation,”
Space Science Reviews 21, 129-157, 1977.
Conor Nixon is a planetary
scientist– at the University of
Maryland, working at
NASA Goddard Space Flight
Center in support of the Cassini
spacecraft mission to Saturn. He is
a co-investigator of the Cassini
Composite Infrared Spectrometer
(CIRS), with responsibilities for
aspects of science planning, data
acquisition and calibration. His research focuses on the
composition of planetary atmospheres, especially measuring
the abundances of hydrocarbons in giant planet
stratospheres to learn about stratospheric circulation, and
the chemistry and evolution of the atmosphere of Saturn’s
moon, Titan. He is also interested in related aspects of
instrumentation and remote sensing. He hails from the UK
where he obtained the degrees of MA in Natural Sciences
(Cambridge University), MSc in Radio Astronomy
(Manchester University), and DPhil in Atmospheric Physics
(Oxford University).
[4] Conor A. Nixon, Nicholas A. Teanby, Simon B. Calcutt,
Shahid Aslam, Donald E. Jennings, Virgil G. Kunde, F.
Michael Flasar, Patrick G. J. Irwin, Fredric W. Taylor,
David A. Glenar and Michael D. Smith, “Infrared limb
sounding of Titan with the Cassini Composite InfraRed
Spectrometer: effects of the mid-IR detector spatial
responses,” Applied Optics 48(10), 1912-1925, 2009.
[5] Nicholas A. Teanby and Patrick G. J. Irwin, “Quantifying
the effect of finite field-of-view size on radiative transfer
calculations of Titan’s limb spectra measured by CassiniCIRS,” Astrophys. Space Sci. 310, 293-305, 2007.
[6] F. Michael Flasar, V.G. Kunde, M.M. Abbas, R.K.
Achterberg, P. Ade, A. Barucci, B. Bézard, G.L. Bjoraker,
J.C. Brasunas, S.B. Calcutt, R. Carlson, C.J. Césarsky,
B.J. Conrath, A. Coradini, R. Courtin, A. Coustenis, S.
Edberg, S. Edgington, C. Ferrari, T. Fouchet, D. Gautier,
P.J. Gierasch, K. Grossman, P. Irwin, D.E. Jennings, E.
Lellouch, A.A. Mamoutkine, A. Marten, J.P. Meyer, C.A.
Nixon, G.S. Orton, T.C. Owen, J.C. Pearl, R. Prangé, F.
Raulin, P.L. Read, P.N. Romani, R.E. Samuelson, M.E.
Segura, M.R. Showalter, A.A. Simon-Miller, M.D. Smith,
J.R. Spencer, L.J. Spilker and F.W. Taylor, “Exploring the
Saturn System in the Thermal Infrared: The Composite
Infrared Spectrometer,'' Space Science Reviews 115, 169297, 2004.
Richard Achterberg is a research
scientist in the Department of
Astronomy at the University of
Maryland, working full time in the
Planetary Systems Laboratory at
NASA's Goddard Space Flight
Center. He is a Co-Investigator on
the Cassini Composite Infrared
Spectrometer (CIRS), for which he
provides
science
planning,
observation design, and instrument operations support. His
research interests include the dynamics of planetary
atmospheres, and the retrieval of atmospheric structure
from thermal infrared data. He has a B.A. in physics and
mathematics from St. Olaf College, and a Ph.D. in
planetary science from the California Institute of
Technology.
[7] Richard K. Achterberg, B.J. Conrath, P.J. Gierasch, F.M.
Flasar, C.A. Nixon: “Titan's middle-atmospheric
temperatures and dynamics observed by the Cassini
Composite Infrared Spectrometer,” Icarus 194, 263-277,
2008.
13
F. Michael Flasar received the
Ph.D.
in
Physics
from
Massachusetts
Institute
of
Technology. He is a research
scientist in the Planetary Systems
Laboratory, NASA/Goddard Space
Flight Center.
His primary
research is the meteorology and
dynamics
of
planetary
atmospheres, including those of
Mars and the outer planets. He was a co-investigator on
the Voyager infrared spectroscopy experiment (IRIS), a
member of the Galileo Radio Propagation team, and a
participating scientist on the Mars Global Surveyor Radio
Science team. He is the principal investigator of the Cassini
Composite Infrared Spectroscopy experiment, and a
member of the Cassini Radio Science team.
14
15