Infrared Limb Sounding With Cassini CIRS
Transcription
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. 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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. 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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