jmaps observations planning simulator - IOPscience

The Astronomical Journal, 149:173 (17pp), 2015 May

doi:10.1088/0004-6256/149/5/173

© 2015. The American Astronomical Society. All rights reserved.

JMAPS OBSERVATIONS PLANNING SIMULATOR Viktor Zubko1, Gregory S. Hennessy2, and Bryan N. Dorland2 1

NOAA Center for Weather and Climate Prediction, College Park, MD 21740, USA; [email protected] 2 United States Naval Observatory, Washington, DC 20392, USA; [email protected], [email protected] Received 2014 February 23; accepted 2015 February 24; published 2015 April 28

ABSTRACT We describe the Joint Milli Arcsecond Pathfinder Survey (JMAPS) Observations Planning Simulator (JOPS), a software simulation and mission planning tool designed to simulate all the observations for a 3 yr, all-sky, spacebased astrometric and photometric survey mission JMAPS. JOPS provides an optimized spatial and temporal distribution of observations given a large number of constraints. The results of our simulations reported here demonstrate the powerful capabilities of the simulator. Key words: astrometry – instrumentation: detectors – methods: numerical We have two objectives in this paper: first, we present the theoretical basis of the JOPS simulator and algorithms; second, we present the initial set of results that demonstrates the capabilities of JOPS. This work is important beyond the JMAPS mission. At its essence, JMAPS was a pointed mission, with multiple constraints; the purpose of JOPS was to optimize the distribution of observations in time and space, maximizing the observing efficiency while enforcing spatial uniformity while adhering to all the constraints. In this sense, the JOPS capabilities are not restricted to just JMAPS. They can be adapted and deployed for use by other astrometric missions (such as the proposed JASMINE) or any pointed mission or ground based observing program seeking to optimize multiple observing parameters while obeying geometric and temporal constraints. The structure of the paper is as follows. In Section 2, we discuss the JMAPS mission in sufficient detail to clearly communicate the nature of the JOPS problem. In Section 3, we describe the major simulator components and their functions: generation of spacecraft orbital and instrument boresight motions, generation of reference sky cells, and the metric tools to quantify observational performance. We present and discuss results from the simulation in Section 4, with conclusory remarks in Section 5.

1. INTRODUCTION The Joint Milli-arcsecond Pathfinder Survey (JMAPS) was a Department of Navy space-based, bright-star, all-sky, astrometric and photometric survey intended to update the current bright star catalogs to 1 milli-arcsecond (mas) accuracy levels at the 2015 epoch. JMAPS was canceled in 2011, but not before a significant amount of progress was made in many areas, including mission concept development. JMAPS was required to observe the entire sky at least 72 times over the course of a 3 yr mission. The basic observing cadence was to point at a specific location, take a twenty second image, then spend ten seconds shifting half a field and settling, followed by another twenty second image, etc. This step-stare approach to observing is fundamentally different from the Hipparcos (Perryman et al. 1997) and Gaia (Perryman 2005) concept of a spinning satellite with apertures that obey a predetermined and relatively fixed-scanning law. There are both advantages and disadvantages to the JMAPS plan. These are discussed in more detail in Zacharias & Dorland (2006). One major challenge this approach creates is that of mission planning. If JMAPS simply scanned up and down in the sky along lines of constant R.A., one of the more simple scan laws, it would heavily oversample the celestial pole regions and undersample the rest of the sky. The challenge with JMAPS was to find an approach that allowed the mission planners to distribute the available observing time evenly over the entire sky such that the mission could meet the 72 observations over 3 yr requirement. Further complicating this problem are additional constraints placed on the observations. These include Solar and Earth exclusion angles, slew times due to the small field shifts (socalled “short slews”) as well as much larger ones due to the orbital motion of the satellite (“long slews”), downlink blackout constraints, and mode switches out of the standard astrometric observations and into other observing modes. In order to demonstrate the feasibility of the overall mission concept as well as begin the development of the actual mission time allocation and planning process, USNO developed the JMAPS Observation Planning Simulator (JOPS). JOPS had reached initial operating capability by the time mission activities ceased in 2012, and was advanced enough to demonstrate mission planning “closure” with all major constraints in place.

2. BRIEF MISSION OVERVIEW The JMAPS mission was developed as an implementation of the new astrometric satellite concept proposed for the first time by Zacharias & Dorland (2006). The JMAPS instrument consists of a small imaging telescope with a single aperture, 19 cm diameter, 3.8 m effective focal length, and single field of view (FOV) of about 1◦. 25 × 1◦. 25 (Dorland & Dudik 2009). When performing astrometric observations, a 2 × 2 set of detectors will be located in the focal plane with each of the detectors being a 4088 × 4088 pixel matrix. Typically, the instrument will observe the sky in a step-stare mode by taking the 20 s exposure followed by the 10 s spacecraft shift (including slew and settle components) by a half-FOV to repeat this sequence again. Note that the single FOV step-stare observing concept is different from the two FOV scanning concept embodied in the ESA Hipparcos mission (Lacroute 1982; ESA 1997). Use of a two-FOV instrument is quite costly because the angles between the two FOVs must be measured to very high accuracy and 1


Zubko, Hennessy, & Dorland Table 1 JMAPS Simulation Parameters

must be monitored constantly. As demonstrated by Zacharias & Dorland (2006), a single FOV approach is capable of achieving 1 mas (or better) astrometry at a significantly lower cost, and the instrument design is simpler in many respects. In addition to doing standard 30 s observations targeting stars in the magnitude range of 6–14 mag, a number of distant Quasi-stellar Objects (QSOs) need be observed in order to provide the zero-parallax reference sources required by the astrometric processing algorithms. Because the QSO sources are considerably fainter than the stellar sources, they require longer exposure times (up to 500 s). On the other hand, very bright objects (0.5–6 mag) need much shorter exposures (down to 0.003 s). It was anticipated that JMAPS should spend about 67% of the mission time in the astrometric mode. The rest of time is spent in other modes: open observations (with several filters covering the range 451–900 nm; to be used for maximum sensitivity searches for faint objects as well as for the multicolor photometry), V-band observations (with a filter approximating a standard Johnson V filter profile in 505–595 nm), spectrophotometric observations (grating in the light pass that creates a dispersed image of the star field in the focal plane covering the spectral range of 451–900 nm), calibration observations (both self-calibration and target oriented) and supporting modes (telemetry downloads/idle/ standby/safe). JMAPS was expected to fly in a 900 km Sun-synchronous polar orbit with the orbital inclination of 99°, orbital period of 103 minutes, near zero eccentricity, and ascending node of 6am (90° from the direction to the Sun). The choice of the orbital parameters results in a “dawn-dusk” orbit with the satellite remaining nominally above the terminator. Such orbit would provide stable thermal and lighting conditions that minimize variations in thermal loads and scattered light backgrounds to the instrument. The anticipated mission duration was 3 yr. The single measurement precision (SMP) can be defined as the expected standard error of a single astrometric measurement using a single exposure. As the JMAPS focal plane simulations show, the value of the SMP is strongly dependent on stellar magnitude and exposure time with the SMP being less than or equal to 7 mas across the useful FOV for all magnitudes 0.5– 12 mag (Dorland et al. 2009). Combining this SMP estimate with the mission requirement that the entire sky should be observed 72 times over the mission life should result in the final 1 mas catalog. Table 1 summarizes the JMAPS mission parameters important for our simulations. These include orbital elements, instrument properties, and main mission constraints.

Parameter

Designation

Value

Orbit L L i e l☉ - lW

Height Period Inclination Eccentricity Ascending Nodea

900 km 103 minutes 99° 0 90°

Instrument L L a

Telescope Diameter Effective Focal Length Field of View Size

Solar Battery Angle

Main Mission Constraints q battery

Earth Limb Angle Field of Regard Angle Max. FOV Envelope Angle Mission Duration Each Object Observability

L

qFOR max qenv L L

19 cm 3.8 m 1◦. 25

20° 40° 40°–55° 11° 3 yr 72 times

Note. a The difference of the ecliptic longitudes of the Sun and ascending node is presented.

includes several components which compute positions on the sky for the spacecraft, telescope boresight, and sensor FOV at the given time t. Then, the observations performance metrics are updated. The loop for non-observing operations just counts time consumed by the operation, and does nothing more. At the next step, preparations for the next operation are done, for example, checking whether the short or long slew is required and in what direction (if applicable). Depending on the situation, the time increment Dt for the starting time of next operation is calculated. Then, for either type of operation, the simulator checks whether the next operation time moment exceeds or is equal to the final simulation time moment tend : when yes, the loop breaks, and the simulator saves the results of calculations in both ASCII and graphical form; otherwise, at the next step, it verifies whether it is time to change operation and its mode (for the observing case), and when yes, it makes necessary changes. Afterward, the loop re-starts again for the new operation. Finally, it is worth noting that the JOPS uses the Gnuplot graphical utility (Phillips 2012) for building plots and LATEX typesetting system (Kottwitz 2011) for creating graphical reports. Below, we present a detailed description of the most important simulator components.

3. JOPS INTERNALS JOPS has been implemented as a C program. Its main components and their interaction are shown on the flow chart in Figure 1. Generally, the JOPS starts its work by reading the input simulation parameters from a configuration file. Then, it generates the sky cell system that tiles the entire sky, and serves as a reference system for quantifying the JMAPS observation performance. After that, the simulatorʼs variables are initialized to the time of the initial observation tstart , the initial observing operation mode along with the associated time parameters are set, and then a simulation loop over time starts. Note that the loop has two branches: one for observing operations and the other for non-observing. The loop for observing operations

3.1. Simulating JMAPS Orbital Motion As noted above, JMAPS was planned to fly on a Sun synchronous orbit. Such orbits are possible due to the asphericity of the Earth which is slightly bulged at the equator. Thus, a near-polar orbit is affected asymmetrically by the bulge gravitation resulting in a slow rotation (precession) of the orbitʼs plane about the axis of the Earth. By choosing the right orbital elements, the orbitʼs precession will match the Sunʼs motion across the sky. Therefore, the orbital plane completes one full rotation about the Earthʼs axis in 1 yr at the rate of about one degree per day. 2


Zubko, Hennessy, & Dorland

Figure 1. Flow chart of the JOPS. Shown are main JOPS components and the order of their execution for the cases of observing operations (in green), non-observing operations (in brown), and other components (in red) as well as input/output operations (in blue). The time moment when the initial observation is to be done is designated as tstart , and the final time of the simulation is tend . The time increment Dt for the next operation may include times for the short or long slewing and settling (when applicable). The time to change operation (observing mode) is denoted as top (tm ), the time period to perform observing (non-observing) operation is Dtobs (Dt nobs ), and the time period to carry out the observing operation in the chosen mode is Dtm . The box for a non-observing operation (like calibration, data transfer to the ground station, internal test, etc.) is drawn by dash line in order to stress that the simulator actually does nothing for such cases except counting time spent while performing the operation.

In our implementation of the JOPS, two spherical coordinate systems are used for describing various JMAPS motions. One of them is the geocentric ecliptic coordinate system shown in Figure 2. It uses the ecliptic, that is the plane of the Earth orbit as a reference plane. The coordinates are given in terms of latitude and longitude. The ecliptic latitude β is measured positive toward the north pole, 0°–90° (and negative toward the south, 0° to −90°). The ecliptic longitude λ is measured eastwards from the vernal equinox point, 0°–360°. Note that the vernal and autumnal equinoxes are located on the line of nodes, the intersection of the ecliptic and celestial equator planes. Another coordinate system we use is based on the instant spacecraft orbit plane: spacecraft orbit coordinates, latitude b and longitude l (Figure 2). Similarly to the ecliptic coordinates, the orbit latitude is measured positive toward the north orbit pole and negative toward the south, whereas the orbit longitude is counted eastward from the orbit ascending node Ω, the intersection of the ecliptic and spacecraft orbit planes. The angle between the ecliptic and spacecraft orbit planes, called inclination i, is equal to 99° for JMAPS. Therefore, because i > 90, the orbit is retrograde: the spacecraft moves in the direction opposite to the Earth rotation. Another important

orbital parameter is the difference between the ecliptic longitudes of the Sun and the orbit ascending node: l Sun - lW. Because of the Sun-synchronicity of the JMAPS orbit, this difference will be kept constant at all times: l Sun - lW = 90 (6 hr in R.A. measure). Thus the orbit coordinate system will gradually precess following the Sun. Note that because the plane of the Earthʼs orbit is fixed in space, the position of the ecliptic plane is fixed too. However, because the Earthʼs rotation axis precesses with a 25,800 yr cycle, so does the celestial equator plane (associated with the axis) and thus both equinoxes (vernal and autumnal). As a result, the equinoxes are sliding westward along the ecliptic with annual motion of 360°/(25,800 yr) ≈ 50.23 yr-1. We do not use this effect here, but, instead, we fix the coordinate epoch to J2000.0. It was anticipated that the detailed planning/ scheduling for JMAPS would be produced by a customized commercial off-the-shelf software product. Such products are capable of modeling the spacecraft motion with great accuracy by taking into account all sizable gravitational and nongravitational perturbations, mission constraints, and observing effects like, e.g., the equinox motion and starlight aberration. The ecliptic coordinates are natural for observation planning and quantifying purposes, whereas the spacecraft orbit 3



Figure 2. Geometry of the Sun synchronous “dawn-dusk” orbit. Two spherical coordinate systems are presented: ecliptic and spacecraft orbital. Ecliptic and respective zero meridian planes are shown in red (bold and thin, respectively). Gridding on the celestial sphere corresponds to the ecliptic coordinates: longitude λ and latitude β with the zero point (λ = 0°, β = 0°) at the vernal equinox (Υ). The instantaneous spacecraft orbit plane and respective zero meridian plane are in blue, and the associated spacecraft orbit coordinates l , b have their zero point (l = 0°, b = 0°) at the orbit ascending node (Ω). The direction of the spacecraft motion is in green.

l W (t ) = l ☉ (t ) - 90 , b W (t ) = 0,

coordinates are ideal for simulating the spacecraft orbital motion and boresight rotations required by the observing modes aligned with the orbital motion. By using the Euler angle technique summarized in Appendix A, the explicit formulas for the transformations from the ecliptic coordinates to spacecraft orbit coordinates and back are derived in Appendix B. Because the JMAPS orbit is circular, the spacecraft moves with the constant angular velocity. Therefore, equations of JMAPS orbital motion in the spacecraft orbit coordinates [ lsc (t ), bsc (t )] are extremely simple: lsc (t ) = lsc0 + wsc ( t - t0 ), bsc (t ) = 0,

where l ☉0 is Sunʼs ecliptic longitude at t0, and w☉ is Sunʼs mean angular velocity along the ecliptic: w☉ = 360 P☉ with P☉ being the period of one Sunʼs full cycle along the ecliptic with respect to the distant stars that is called the sidereal year: P☉ = 365.2564 days. Thus, it is easy to derive: w☉ » 59′.14/days. Note, however, that the Earth’ orbit is not an exact circle, but instead has a small ellipticity. As a result, the Sunʼs angular velocity shows small variations along the orbit, which are within a few percent. We assume in Equations (2), (3) that the spacecraft orbit precession velocity matches the Sunʼs mean velocity w☉. After substituting Equations (1)–(3) in the coordinate transformation Equations (B6a) and (B6b) and performing some trigonometric manipulations, we arrive at the final expressions:

(1)

where t is the running time, t0 is some initial time, lsc0 is the spacecraft longitude at t0, and wsc is the spacecraft orbital angular velocity, which is equal to wsc = 360 Psc . Substituting the orbital period for JMAPS, Psc = 103 minutes, in the last formula results in wsc » 3′.5 s−1. To transform the coordinates (1) to the ecliptic form, the motion of the Sun and that of the orbit ascending node should be defined in the ecliptic coordinates (l ☉ (t ), b☉ (t ), and lW (t ), bW (t ), respectively): l ☉ (t ) = l ☉0 + w☉ ( t - t0 ), b ☉ (t ) = 0,

(3)

l sc (t ) = arctan2 éë -cos lsc (t )cos l ☉ (t )

+ sin lsc (t )sin l ☉ (t )cos i , cos lsc (t )sin l ☉ (t ) + sin lsc (t ) ´ cos l ☉ (t )cos iùû,

(2)

4

(4a)


bsc (t ) = arcsin [ sin i sin lsc (t )] ,


each. Each observing block is ≈57° in length and covers 99 exposures. After completing an orbit, the boresight is offset by 1/2 FOV perpendicularly to the orbital motion (along the orbit latitude b) to optimize the coverage: b btnew = b btold + s · a 2 and l btnew = l btold , where the sign s = ±1 depends on the previous boresight position. When the boresight envelope angle qenv limit is reached, the sign of the offset is changed to the opposite, s  -s , and the progression of the scan over the orbit latitude reverses. Once the boresight direction (l bt, b bt ) is set, the FOV can be defined through its vertices shown in Figure 4. In the spacecraft orbit coordinates, the FOV vertices are:

(4b)

where function arctan2(y, x) is defined in Appendix B. 3.2. Simulating Astrometric Observations The astrometric observing is conducted in one of three basic scanning modes: in-scan, cross-scan, and anti-Sun. In all three modes, the instrument boresight will be commanded to point to a desired location on the sky defined by the ecliptic coordinates and the roll angle: a slew and settle operation will occur, followed by an exposure of specified length. After the exposure is completed, the spacecraft will slew to the location of the next exposure with a typical offset of 1/2 FOV (0◦. 625), settle and expose, and then the process repeats until the observational constraints are encountered. A cone about the zenith direction defines the acceptable field of regard (FOR) of the instrument boresight, within which observing is allowed. The respective FOR zenith angle qFOR should meet the Earth-limb scattered light limit of 40°. Taking into account that at a 900 km orbit, the Earth subtends a half angle of about 62°, the maximum possible FOR angle is 78°. The specific value of qFOR , typically within the range of 40-55, will be chosen to maximize the observing efficiency.

lFOV,1 = lbt - a 2, bFOV,1 = b bt + a 2,

(5a)

lFOV,2 = lbt + a 2, bFOV,2 = b bt + a 2,

(5b)

lFOV,3 = lbt + a 2, bFOV,3 = b bt - a 2,

(5c)

lFOV,4 = lbt - a 2, bFOV,4 = b bt - a 2.

(5d)

The spacecraft orbital coordinates of the boresight and FOV, l bt, b bt , and l FOV, i, b FOV, i , i = 1..4 can be transformed into the ecliptic coordinates l bt , b bt , and l FOV, i, b FOV, i , i = 1..4 by applying Equations (B6a) and (B6b). As can be seen in Figure 4, the orientation of the FOV is defined by the roll angle qroll , the angle between the normal to the solar battery plane and the ecliptic meridian that passes through the boresight point on the sky:

3.2.1. In-scan Observing

In the in-scan observing mode, the instrument boresight is pointed at 90  qbattery to the Sun–Earth direction, and is advanced after each exposure by 1/2 FOV in the direction of orbital motion. The solar battery angle qbattery = 20° is an important mission constraint imposed on the boresight motion to keep the spacecraft battery in properly charged at all times. The JMAPS spacecraft was designed to have the solar panels produce enough power for the spacecraft without needing battery power as long as the solar panels were pointed to within qbattery of the Sun. Figure 3 shows the boresight envelope (green lines) defined by the solar battery constraint, which limits the boresight motion max within qenv = qbattery - i + 90 = 11 from the spacecraft orbit plane. Note that 11° is the maximum possible angle of the envelope, and the actual qenv may be set to smaller values to optimize the mission efficiency. When l bt and b bt are the instrument boresight longitude and latitude in the spacecraft orbit max should hold for coordinate system, then the relation ∣ b bt ∣ ⩽ qenv the in-scan observing at all times, and the boresight scanning advancement can be written as: l btnew = l btold + a 2 and b btnew = b btold where a is the FOV size (a = 1◦. 25) and “new” and “old” refer to the new and previous boresight positions, respectively. The rate of the in-scan boresight advancement is ≈1◦. 25 minutes−1. However, the spacecraft orbital motion is effectively moving the boresight with respect to the zenith by 3◦. 6 minutes−1. Thus the scanning slew is not fast enough to keep the boresight within the FOR continuously. When the boresight hits the trailing FOR limit (-qFOR ), the large “recovery” slew is executed, during which the boresight rotates in the direction of spacecraft orbital motion past the zenith to the leading FOR limit (qFOR ), and then the standard exposureslew-settle sequence runs again. For example, as our simulations show, the operational efficiency can be increased when the boresight is pointed in half-orbit blocks. When qFOR » 52, for a nominal 103 minutes orbit, there are two blocks of ≈49.5 minutes and two recovery slews of about 2 minutes

tan q roll = ( -cos lbt sin i ) æ çç ( cos b bt cos i - sin b bt sin lbt sin i ) çè æ 2 öö ´ çç 1 - ( sin b bt cos i + cos b bt sin lbt sin i ) ÷÷÷÷÷ . (6) ÷øø çè

The derivation of this equation is presented in Appendix D. Equation (6) uses the spacecraft orbit coordinates of the instrument boresight, thus resulting to the simplest possible expression for qroll . When desired, qroll may be expressed through the ecliptic coordinates by using the coordinate transformation Equations (B6a) and (B6b), but the final expression will be more complicated. For the FOV and solar battery orientation adopted above, the angle qnormal between the normal line to the battery plane and the Earth–Sun line can be expressed using the spherical law of cosines: cos qnormal = cos b bt sin i + sin b bt cos i sin lbt .

(7)

It is easy to show that for any l bt , the angle qnormal is within i - 90  ∣ b bt ∣ with the limit values corresponding to l bt = 90° and 270° (as seen in Figure 3). This ensures that the battery max constraint ∣ b bt ∣ ⩽ qenv is met for any boresight direction within the allowable envelope. 3.2.2. Alternate Modes of Observing

Since only in-scan observing was simulated in the current process, we describe the cross-scan and anti-Sun observing in Appendix C. 5



Figure 3. Geometry of the angles important for astrometric in-scan and cross-scan observations. An edge-on view of the spacecraft orbit plane from within the ecliptic plane. Gridding corresponds to the ecliptic coordinates with the ecliptic plane in red. The normal to the ecliptic plane is in black. Instant spacecraft orbit is in blue, and the limits of the maximum instrument boresight envelope are in green. The magenta and cyan straight lines indicate the maximum possible instrument boresight deviation from the spacecraft orbit plane consistent with the solar battery constraint.

1. The celestial sphere is decomposed into spherical “linear” quadrangles. The decomposition is non-hierarchical because there is no need in this feature in our simulator. 2. For each sky cell, two cell sides are located along constant latitudes with the other two oriented along constant longitudes. In addition, the cells are aligned along lines of constant latitude. Therefore, the computational algorithms for building and using such the cell system are quite simple. 3. Cells are of approximately equal area.

3.3. Reference Sky Cells For the purposes of planning of JMAPS observations, the simulator needs a system of reference sky cells, which allow one to estimate the observability of various sky areas over extended time periods up to the anticipated mission life time. The sky cells should cover the whole sky, be defined in ecliptic coordinates, and be quite simple for quantifying observations. Of wide use in astronomy is the sky subdivision scheme implemented in the Hierarchical Equal Area isoLatitude Pixelization of a sphere (HEALPix) software package (Górski et al. 2005). The aim of HEALPix is to provide a base for harmonic analysis of sky radiation at various wavelengths measured by cosmic missions like COBE, MAP, and future Planck. The main features of HEALPix are: (1) the sphere is hierarchically divided into curvilinear quadrangles, (2) areas of all cells are identical, and (3) cell centers are distributed on lines of constant latitude. The identity of cell areas in HEALPix comes for the price of having curvilinear cells. We found that for the JOPS simulator purposes, it is desirable to have simpler, “linear” quadrangular cells aligned with the ecliptic coordinate lines. Thus, as a result of our numerical experiments with JOPS, we have created a sky cell system with the following properties.

To generate the cell system with the above properties, just one parameter needs to be defined: cell side size c (in arc degrees), which, in turn, defines the desired cell area c2 (in arc degrees squared, respectively). Then, a sequence of formulas for deriving grid points over the latitude is as follows: é 180 ù h b0 = c  n bgrid = 1 + êê 0 úú êë h b úû 180  h b = grid nb - 1 ↪b igrid = -90 + h b (i - 1), i = 1, 2 ,..., n bgrid ,

6

(8a)

(8b)



Figure 4. Schematic representation of the instrument FOV orientation in astrometric in-scan and cross-scan modes. The FOV is seeing face-on. The focal plane that contains the FOV is perpendicular to the solar battery plane. The normal to the battery plane characterizes the orientation of the FOV which is defined by the roll angle q roll . Shown are also the instrument boresight and the four vertices of the FOV (numbered from 1 to 4). The left and right parts of the figure present the FOV orientation in the spacecraft orbit and ecliptic coordinates, respectively.

where h b0 is the initial step over the latitude β which is set to the cell size c, n bgrid is the number of points over β, [·] is a function which retrieves the integer part of a real number, h b is the

sequence of formulas (i = 1, 2,..., n bgrid - 1): Si0 = c2  hl0,i =

adjusted step over β, and bigrid are grid points over the latitude with i running from 1 to n bgrid . To move further with calculating grid points over the longitude, we need the expression for the area of the quadrangular cell. When its sides are oriented along the latitude and longitude isolines: from b1 to b 2 and from l1 to l 2 , respectively, then, in the ecliptic coordinates, the area is a simple integral: S ( b1, b 2, l1, l 2 ) = =

l2

b2

1

1

òl òb

grid sin b igrid + 1 - sin b i

é 360 ù ê ú nlgrid ,i = 1 + ê 0 ú h êë l,i úû 360 hl, i = grid nl,i - 1 ↪ Si =

cos b db dl

1 ( l2 - l1)( sin b2 - sin b1), f

fSi0

hl, i f

( sin b

grid i+ 1



(10a)

)

- sin b igrid 

l igrid ,j = hl, i (j - 1) , j = 1, 2,..., nlgrid ,i - 1,

(9)

(10b)

where S0i is the initial cell area equal to the target cell area c2, hl0,i is the initial step over the longitude λ, nlgrid ,i is the number of points over λ, hl, i is the adjusted step over λ, Si is the adjusted cell area based on the adjusted step, and l igrid are grid ,j

where S is the area square (in square degrees), longitude λ and latitude β are in degrees, and f is a converting factor from degrees to radians, f = p 180. Note that in the limiting case of b1 = -90, b 2 = 90, l1 = 0, and l 2 = 360, the quadrangle covers the whole celestial sphere, and actually degenerates in the spherical lune. As follows from Equation (9), the area in this case is S = 4π/f2 sq. degrees, that is 4π steradians (as expected). The grid points over the longitude are latitude dependent now (through index i). They can be derived with the following

points over the longitude with j running from 1 to nlgrid ,i - 1. Using the derived grids over λ and β, we can construct a cell system covering the whole sky. Each cell is characterized uniquely by the pair of indices: i (latitude, from 1 to n b = n bgrid - 1) and j (longitude, from 1 to nl, i = nlgrid ,i - 1) and has cell vertices: (l i, j , bi ), (l i, j , bi + 1), (l i, j + 1, bi + 1), and (l i, j + 1, bi ), and cell center coordinates: l ic,j = (l i, j + l i, j + 1 ) 2

7



Figure 5. Sky cell centers are shown on the celestial sphere for the case when the cell side size c = 1°. The number of cell center points over the latitude and total number of cells are shown. Table 2 Sky Cell Systems

nb

c 1◦. 000 0◦. 300 0◦. 100 0◦. 030 0◦. 015

180 600 1800 6000 12,000

nlmin 4 4 4 4 4

a

nlmax

n cells

min b n FOV

max b n FOV

min c aFOV

max c aFOV

41,344 458,654 4,126,190 45,839,628 183,352,438

0 4 103 1240 5094

1 16 144 1681 6889

0.0% 23.0% 65.9% 71.4% 73.4%

64.0% 92.2% 92.2% 96.8% 99.2%

a

360 1200 3600 12,000 24,000

Notes. a Minimum/maximum number of cells over the longitude: nlmin = min1 ⩽ i ⩽ nb nl, i , nlmax = max1 ⩽ i ⩽ nb nl, i . b c

Minimum/maximum number of entire cells in the FOV for astrometric observations. Minimum/maximum percentage of the FOV area occupied by entire cells.

and bic = (bi + bi + 1 ) 2. Note that the cells that have a pole as a vertex (i = 1 for the south pole, and i = n b for the north pole) degenerate in the triangles. The total number of sky cells over the sphere is n cells = å in=b 1 nl,i . Figure 5 shows an example of the sky cell system for the case of cell size c = 1°. Note, however, that JOPS simulations were usually done with smaller cell sizes down to c = 0◦. 015. The parameters of some typical sky cell systems are listed in Table 2.

some sky area, the FOV contains some amount of full sky cells and a smaller amount of partial sky cells (mostly along the FOV boundary). The smaller the cell size c the larger amount of full sky cells are seen in the FOV. Every cell on the sky can be assigned the number of times the cell is seen in the FOV during a specified time period. We call it the number of FOV hits per cell Hi, j , where i = 1 .. n b , j = 1 .. nl, i . Obviously, Hi, j characterizes the two-dimensional (latitude and longitude dependent) angular distribution of the number of FOV hits over the celestial sphere. Then the mission requirement that every sky cell (i,j) should be observed 72 or more times during the mission lifetime can be written as Hi, j (t ⩾ 3 yr) ⩾ 72. Mathematically, computation of Hi, j reduces to the problem of finding whether a sky cell is inside the sensorʼs FOV. A detailed description of the algorithm for solving this problem is presented in Appendix E.

3.4. Metrics to Quantify JMAPS Observations Performance Once the reference sky cell system is defined, we can introduce quantifiers of JMAPS observations performance. Two parameters are most important here: the cell size c and FOV size a. For the quantifying to make sense, c should be smaller than a: c < a . When JMAPS makes an exposure of 8



Figure 6. Example of the weekly dependence of the number of sky cell observations normalized by the total number of sky cells. Shown are the partial, cumulative, and total numbers. The re-scaled partial number with the ticks to the right is presented for better clarity.

Because the orbit of JMAPS was expected to be near polar, a strong dependence of Hi, j on the latitude is expected. Thus, it is useful to derive the averages of Hi, j over the longitude that would be one-dimensional, latitude dependent only numbers of FOV hits per cell as: Hi,min = min Hi, j , 1 ⩽ j ⩽ n l, i

Hi,max = max Hi, j , Hi,mean =

1 ⩽ j ⩽ n l, i n l, i

1 nl, i

å Hi,j, i = 1 .. n b ,

(11)

j=1

where Hi,min , Hi,max , and Hi,mean are Hi, j ʼs minimum, maximum, and arithmetic mean, respectively, over the longitudes at the latitude i. Another useful quantity is the distribution function of the number of FOV hits per cell Ncells(H), which is actually the number of cells with the specified value of H over the celestial sphere. Examples of plots demonstrating the behavior of the above metrical quantities will be presented in Section 4. Finally, as the simulation progresses, JOPS compiles a yearly plan of the numbers of sky cell observations (both partial and cumulative) as well as the detailed list of observability of sky cells as a function of the yearʼs week (for 1 yr simulations only). Both the plan and list can be useful as input for performing more accurate planning and scheduling of JMAPS observations. Figure 6 presents a typical example of the weekly dependence of the number of sky cell observations normalized by the total number of sky cells. Depicted are the partial number which characterizes the total number of sky cell observations over the celestial sphere for a specific week, the cumulative number which is a sum of the partial numbers for all previous and current weeks, and the total number for the whole year which is the natural limit the cumulative number

Figure 7. Fragment of the list of sky cell observability for 1 yr.

goes to in the last week of the year. Figure 7 presents a fragment of a typical list of sky cell observability covering 1 yr. For each week, it presents the list of sky cells (through indices of their longitude and latitude) observable during the specified week as well as the number of times the cell is observable within the week. 4. RESULTS OF JMAPS SIMULATIONS As was mentioned above, the purpose of the development of the JOPS simulator was to create a tool for exploring JMAPS observations planning strategies. By the time the decision to close the JMAPS project had come, the work on implementing the JOPS minimum capabilities listed in Section 3 had mostly been completed. Initially, we planned to perform a comprehensive search for JMAPS optimal observing strategies, once the JOPS was ready. But the project closure and shortage of time forced us to significantly reduce the scope of the investigation. Thus we decided to explore a simple class of 9



observing strategies based on using the in-scan only astrometric mode, which is the main and most efficient observing mode. The aims of the study are as follows: (1) demonstrating JOPS general simulation capabilities; (2) uncovering potential problems in getting an optimal strategy; and (3) finding efficient solutions resulting in the optimal strategy. The results of the study are presented in this section. We first describe the simulation steps common for all our simulation models. Usually, the simulation starts by choosing the sky cell system. It is preferable to use smaller sky cells because, in this case, the fraction of the FOV area occupied by entire cells, which contribute to calculating the number of FOV hits per cell H, is higher. In all our reported simulations, we used the smallest possible cell size c equal to 0◦. 015, which is only limited by the existing memory on our computer (16Gb). At the next step, the optimal FOR angle, qFOR , is selected by running simulations for various qFOR and picking that qFOR which maximizes the percentage of sky observed 72 and more times. The resulting qFOR is around 52°, and this value is used for all our simulations. A starting case of our JOPS simulations that we call the initial observing strategy or Strategy 0, was the 3 yr observing period with 67% of the time spent in the in-scan observations including short and long slews, and 33% of the time reserved for other works such as, e.g., data down-linking, calibration, or observations in other modes. More precisely, the time control of operations is done on a daily basis: 16 (contiguous) hours to be spent for in-scan observing and 8 hr for other operations. In the JOPS Flow Chart (Figure 1), it means that the initial obseving mode is set to “Astrometric In-scan,” and the time period for this operation is set to Dtobs = 16 hr. In the box “Time to Change Observing Mode?” the selection will always be “No,” because the only mode used in the present case is “InScan.” The loop over the observing operation runs until the current time t in box “Time to Change Operation?” exceeds or is equal to top . Then, the operation is changed to “Nonobserving” with the time period for the operation to be set to Dtnobs = 8 hr, and the loop over the non-observing operation runs one time (we choose Dt = Dtnobs = 8 hr for this). Then, after passing box “Time to Change Operation?,” the operation switches back to “Astrometric In-scan,” and the simulation algorithm re-starts again until t exceeds or is equal to tend . A more complicated simulation would assign a non-zero time allocation for mode simulated. The total time allocation fraction needs to sum to unity. No other constraints in addition to the solar battery, Earth limb and FOR are used. The instrument boresight envelope is set to its maximum: max qenv = qenv = 11. Figure 8 presents the results of our calculations. It can be seen in Figure 8 that there are three distinct types of sky regions for the number of FOV hits per cell H depending on the latitude.

Figure 8. Results of JOPS simulations for the 3 yr period: 67% of time spent in astrometric in-scan mode. Strategy 0: the boresight envelope angle is set to its maximum, qenv = 11, and no constraints on the boresight latitude. Shown are: (a) number of FOV hits per cell, H on the celestial sphere; (b) H minimum, H maximum, and arithmetic mean of H over the longitude as a function of the latitude; and (c) distribution function of H over the celestial sphere and the Gaussian fit to it.

1. The main, non-polar region with ∣ b ∣ ⩽ 70 where H generally goes up with increasing β from the equator (β = 0°). Note that the rate of the H increase becomes steeper when approaching β = 70°. As we found, the arithmetic mean of H over the longitude, Hmean , can be approximated very well by the following simple function: c2

Hfit (b ) = c1 + 4

cos ( b b1)

where c1, c2, and b1 are parameters which are found by fitting Hmean for ∣ b ∣ ⩽ 70. 2. Polar regions with the latitude β between 70° and »87-88. At these latitudes, the H increase with β becomes gradual.

(12)

10



3. Near-polar regions located within couple of degrees from each of the poles, where H shows quite sharp maximums. The behavior of H as a function of the latitude can be understood when we look at the geometry of angles shown in Figure 3. The maximum deviation of the boresight envelope from the poles is defined by the solar battery critical angle qbattery = 20 (angle between the magenta and black lines). It means that the sky areas with highest latitudes ∣ b ∣ ⩾ 70 will be observable almost every “boresight orbit” (a projection of one cycle of in-scan boresight motion on the celestial sphere): the higher the latitude the more sky cells will be exposed along the track with the maximum corresponding to ∣ b ∣ = 90. With the latitude lowering from ∣ b ∣ = 70 toward the equator, the FOV exposition time for the sky cells located at the latitude decreases, and this effect can be seen in Figure 8. Thus, the solar battery constraint results in the separation of polar and non-polar regions with the critical latitude of 70°. The other edge of the boresight envelope is at just 2° from the poles (angle between black and cyan lines). Because it has the longitude opposite to that of the 20°-boresight-edge, the regions within 2° from the poles will be exposed roughly two times more frequently (more FOV hits per cell) than the neighboring areas, and this is why the sharp peak in H is seen there. The distribution function for H, Ncells (H ) presented in Figure 8(c) exhibits four major peaks. The major peaks ordered from high to low are: (1) at H = 56 due to the lower-latitude non-polar region, ∣ b ∣ < 40 (includes most of sky cells); (2) at H = 100 from the higher latitude non-polar region, 60 < ∣ b ∣ < 70; (3) at H = 220 due to the polar region, 70 ⩽ ∣ b ∣ < 87; and (4) at H = 375 originating from the near-polar region, 87 ⩽ ∣ b ∣ < 90. The main conclusion following from Figure 8 is that the polar areas at the latitudes ⩾70° are significantly over-observed. As a consequence, we may see the under-observance of the rest of sky with the total of ≈28% only of the whole sky observable 72+ times. Thus, there is a need to explore other observing strategies that would result in a more homogeneous distribution of the number of FOV hits H over the celestial sphere. We expect that it would also increase the percentage of the sky the can be exposed the required 72+ times. In the next case, called Strategy 1, we make the first step toward the homogenization of the H distribution over the sky: we need to fix the problem of the double exposing of the nearpolar areas. To proceed with it, the instrument boresight envelope angle is reduced to qenv = 9. Then, the edges of the boresight envelope will deviate from the poles by 18° and 0° instead of 20° and −2°, respectively, for Strategy 0. Similarly to the Strategy 0 case, the observing period is 3 yr with 67% of the time spent in the in-scan observations. The results of JOPS simulations for the Strategy 1 case are shown in Figure 9. Now, the polar regions, 72 ⩽ ∣ b ∣ ⩽ 90 show almost constant mean H, whereas H in the non-polar region exhibits same behavior as in the Strategy 0 case. The distribution function Ncells (H ) has just two major peaks at 58 (non-polar cells) and 240 (polar cells). Now, one third of the sky, 34% is observable 72+ times, but, certainly, it is still not enough. We have performed extensive numerical experiments toward finding observing strategies that produce a more homogeneous sky distribution of the number of FOV hits per cell H. As follows from the results for Strategies 0 and 1, the main problem is how to redistribute the observing time from higher

Figure 9. Results of JOPS simulations for the 3 yr period: 67% of time spent in astrometric in-scan mode. Strategy 1: reduced boresight envelope angle qenv = 9, and no boresight latitude constraints. Shown are: (a) number of FOV hits per cell, H on the celestial sphere; (b) H minimum, H maximum, and arithmetic mean of H over the longitude as a function of the latitude; and (c) distribution function of H over the celestial sphere and the Gaussian fit to it.

to lower latitudes. To solve the problem, we employed the boresight latitude constraint that would restrict the highest possible observing latitude to some critical value of b bt in such a way that ∣ b bt ∣ ⩽ b bt < 90. Note that we used the same 11



boresight envelope, qenv = 9 as in Strategy 1. We experimented with various approaches to choose b bt , and found that the following equation yields the best results: 1

   b bt = éê ( b min - b min ) p + ( b max ) p rrng ùûú p , ë

(13)

where the critical latitude b bt is selected randomly once per time period of t  , rrng is the random number generator that produces a real number uniformly distributed between 0 and 1,   (bmax ) is some minimum (maximum) latitude of the b min interval from which choose the critical latitude, and p is some exponential parameter. The time parameter t  was set to one   orbital period (=103 m), and the other parameters, b min , , bmax and p were found by maximizing the percentage of sky observable 72+ times. We will call the simulation approach that uses the observing latitude constraint based on Equation (13) as Strategy 2. Again, we performed our simulations for the 3 yr period with 67% of the time spent in the in-scan observations. In yr 1, the boresight latitude was unconstrained: in order to cover the polar areas. But, in Years 2 and 3, the latitude constraint was applied: as a result, most of observing time was allocated for the non-polar region. The resulting optimum  values of the parameters in Equation (13) are: b min = 57,  bmax = 85, and p = 1.015. The number of FOV hits per cell, H, and its distribution function, Ncells (H ) for the Strategy 2 case are presented in Figure 10. It can be immediately seen that the distribution of cell observations H over the sky is now much more uniform than that derived with Strategies 0 or 1, and it results in about 63% of sky observable 72+ times. The angular dependence of H looks quite good despite using a quite simple approach. It can be further improved by using more sophisticated techniques. The distribution function Ncells (H ) shows just one peak at H = 75. The area around the peak is well approximated by a Gaussian function. Thus, based on our current simulations, a 3 yr mission will observe ≈63% of the sky 72+ times. As a final case, we have used Strategy 2 for extending JOPS simulations to 4 yr aimed at further increasing the percentage of the “72+ times” sky observability. The case is similar to the 3 yr case, with the boresight latitude constraint used in the 3rd year continued through the newly added 4th year. Our simulations have shown that the optimum values of the Equation (13) parameters for the   4 yr case are: b min = 85, and p = 0.81. The = 58, bmax resulting H and Ncells (H ) are depicted in Figure 11. As can be seen, the angular dependence of H over the sky is quite similar to that for the 3 yr case except that, generally, the H numbers are larger by a factor of about 4 3, but this is what is expected when going from 3 to 4 yr mission. The H distribution function has a single maximum at H = 100, which is also well fit with a Gaussian function. Now, 99.46% of sky is observable 72 or more times. Thus, extending the JMAPS mission to 4 yr would definitively ensure that essentially the entire sky can be observed the required number of times. A summary of the above simulation cases is presented in Table 3.

Figure 10. Results of JOPS simulations for the 3 yr period: 67% of time spent in astrometric in-scan mode. Strategy 2: reduced boresight envelope angle qenv = 9, no boresight latitude constraints in Year 1, and latitude constraints are applied in Years 2–3. Shown are: (a) number of FOV hits per cell, H on the celestial sphere; (b) H minimum, H maximum, and arithmetic mean of H over the longitude as a function of the latitude; and (c) distribution function of H over the celestial sphere and the Gaussian fit to it.

observing strategies to get most from the mission. Despite the cancellation of the JMAPS mission, the experience gained while working on JOPS can be useful for planning of future astrometric missions similar to JMAPS, such as JASMINE.

5. CONCLUSIONS 1. The paper presents a comprehensive description of the JOPS, developed for exploring JMAPS optimal 12


Zubko, Hennessy, & Dorland Table 3 JMAPS Simulation Summary Strategy 0 1 2 2

Years

qenv

Lat. Cons.a

P72 +b

3 3 3 4

11° 9° 9° 9°

Off Off On On

28.5% 34.1% 62.9% 99.5%

CPU Timec 8h 8h 6h 7h

59 m 45 m 02 m 49 m

Notes. Latitude constraints applied (On) or not (Off). b Percentage of sky observable 72 and more times. c For Dell XPS 8300, Intel Core i7–2600 3.4 GHz, 16 GB. a

its maximum (qenv = 11°) show that the polar areas with latitudes higher than 60 are significantly over-observed, and the rest of sky is under-observed. As a result, only 28% of sky can be observed the required 72 or more times. 3. Having set a goal to redistribute the observing time from polar to non-polar areas, we reduced the boresight envelope to qenv = 9° and introduced the upper limit for the boresight observing latitude. In application to the 3 yr mission, the new constraints have resulted in a much more uniform distribution of JMAPS observations over the sky with 63% of sky observable 72+ times. It may be sufficient to meet the mission accuracy requirement without additional observations—additional analysis is required. 4. Extending the mission simulations to 4 yr demonstrates that almost the entire sky (>99%) is observable the required number of times.

APPENDIX A EULER ANGLES AND COORDINATE TRANSFORMATIONS The transformation from a three-dimensional (3D) Cartesian coordinate system to another one can be described uniquely by using three rotation angles. However, the choice of the set of such angles is not unique. Of many possibilities, we have chosen a set of angles Φ, Θ, and Ψ as shown on Figure 12. The angles are named the Euler angles and are widely used in astronomy and theoretical mechanics (Goldstein et al. 2001). If axes X and Y lie in the ecliptic plane, then angle Φ is named the precession angle, Θ is the nutation angle, and Ψ is the intrinsic rotation angle. The angle ranges are 0 ⩽ F < 360, 0 ⩽ Y < 360, and 0 ⩽ Q ⩽ 180. Using the Euler angles, the transformation from the Cartesian system (XYZ) to system ( X ¢Y ¢Z ¢) is done in three steps.

Figure 11. Results of JOPS simulations for the 4 yr period: 67% of time spent in astrometric in-scan mode. Strategy 2: reduced boresight envelope angle qenv = 9, no boresight latitude constraints in Year 1, and latitude constraints are applied in Years 2–4. Shown are: (a) number of FOV hits per cell, H on the celestial sphere; (b) H minimum, H maximum, and arithmetic mean of H over the longitude as a function of the latitude; and (c) distribution function of H over the celestial sphere and the Gaussian fit to it.

1. Rotate XYZ system about Z axis by the angle Φ counterclockwise. 2. Take the resultant coordinate system that has its X axis aligned along the line of nodes OW, and rotate it about the line of nodes OW counterclockwise by the angle Θ. Now the Z′ axis is in place. 3. Finally, rotate the coordinate system derived at the previous step about Z′ axis by angle Ψ counterclockwise to produce the desired X ¢Y ¢Z ¢ system of axes.

2. Our initial simulations of the in-scan astrometric observations for the 3 yr mission case when only standard mission constraints—solar battery, Earth limb, and FOR —are imposed and boresight observing envelope is set to 13



Figure 12. Cartesian coordinates systems XYZ and X ¢Y ¢Z ¢ , planes XY and X ¢Y ¢ , and three Euler angles Φ, Θ, and Ψ used for the transformations from XYZ to X ¢Y ¢Z ¢ . O is the center of the both coordinate systems, and Ω is the ascending node.

The inverse transformation from X ¢Y ¢Z ¢ to XYZ can be done similarly by applying the rotations in the following order: -Y , -Q, and -F. Thus, each of the transformations utilizes two rotations about Z axis and one about X axis. Mathematically, a rotation about X, Y, and Z axes by some angle α can be expressed through respective rotation matrices R X (a), RY (a), R Z (a) that have relatively simple form: é1 0 0 ù R X (a) = êê 0 cos a sin a úú, êë 0 -sin a cos a úû

(A1a)

é cos a 0 -sin a ù ú RY (a) = êê 0 1 0 ú, êë sin a 0 cos a úû

(A1b)

é cos a sin a 0 ù R Z (a) = êê -sin a cos a 0 úú. êë 0 0 1 úû

(A1c)

APPENDIX B COORDINATE TRANSFORMATIONS FROM ECLIPTIC TO SPACECRAFT ORBIT AND BACK Here, we apply the rotation matrix (A2) derived in Appendix A to find the relationships between the ecliptic and spacecraft orbit coordinates defined in Section 3.1. The XYZ Cartesian coordinates can be associated with the ecliptic spherical coordinates l , b by assigning the XY plane with the ecliptic plane in such a way that the X axis goes through the vernal equinox Υ (Figure 2). Then the Z axis points to the North Ecliptic Pole. A similar association can be done for the X ¢Y ¢Z ¢ coordinates with the spacecraft orbit coordinates l, b: the X ¢Y ¢ plane is in the spacecraft orbit plane so that the X′ axis points to the orbit ascending node Ω. Then the Z′ axis goes through the North Orbit Pole (Figure 2). Following the above definitions and supposing the formal radius of the celestial sphere to be equal to 1, the relations between the Cartesian and spherical coordinates on the celestial sphere are: é x ù éê cos b cos l ùú ê y ú = ê cos b sin l ú, ê zú ê ë û êë sin b úúû

By using matrices (A1a)–(A1c), it is possible to compute a matrix describing any rotation in 3D space. Thus, the rotation matrix for the three-step transformation from XYZ to X ¢Y ¢Z ¢ (Figure 12) can be expressed as the product of the three matrices corresponding for the separate rotations:

é x ¢ ù é cos b cos l ù ê ú ê ú ê y¢ ú = ê cos b sin l ú. ê ú ê sin b ú û êë z¢ úû ë

(B1)

For the transformation from XYZ (ecliptic) to X ¢Y ¢Z ¢ (spacecraft orbit), the respective rotation matrix REC  SO is calculated from matrix (A2) with the following Euler angles: F = lW, Q = i , and Y = 0. The result is

R (F , Q , Y) = R Z (Y) · R X (Q) · R Z (F) = ù é sin F cos Y cos F cos Y sin Y sin Q ú ê ú ê - sin F sin Y cos Q + cos F sin Y cos Q ú. ê sin sin F Y cos sin F Y ú ê ê - sin F cos Y cos Q + cos F cos Y cos Q cos Y sin Qú ú ê êë sin F sin Q cos Q úû -cos F sin Q (A2)

R EC  SO

é cos l W sin l W 0 ù ê ú = ê -sin l W cos i cos l W cos i sin i ú. ê ú êë sin l W sin i -cos l W sin i cos i úû

(B2)

In a similar manner, the transformation from X ¢Y ¢Z ¢ (spacecraft orbit) to XYZ (ecliptic) requires the Euler angles F = 0 , Q = -i , and Y = -lW. Substituting these values of Euler angles in the matrix R (A2) results in the following rotation

14



matrix RSO  EC : é cos l W -sin l W cos i sin l W sin i ù ê ú RSO  EC = ê sin l W cos l W cos i -cos l W sin i ú. ê ú sin i cos i úû ëê 0

APPENDIX C ALTERNATE MODES OF OBSERVING C.1 Cross-scan Observing

(B3)

The cross-scan observing strategy is similar to in-scan except that the instrument boresight advances in the direction perpendicular to the spacecraft orbital motion, while maintaining the solar battery, FOR and other constraints. The sequence of astrometric observations, overlapping FOV, and recovery slews are similar to the in-scan case. The purpose of cross-scan observing is to break systematic correlations between in-scan stripes. In general, the cross-scan mode requires more recovery slews and is thus less efficient than in-scan. Cross-scan observing will be used in concert with, rather than in exclusion to, in-scan observing. max The maximum boresight envelope with qenv = 11, shown in Figure 3, is also valid for the cross-scan observations. Likewise to the in-scan case, the actual envelope angle qenv can be chosen from the mission efficiency considerations. In the spacecraft orbit coordinates, the cross-scan boresight advancement is: l btnew = l btold and b btnew = b btold + s · a 2, where the sign s = 1. For example, the consecutive exposures move the boresight toward the direction of the Sun, l btnew = l btold and b btnew = b btold - a 2, until the qenv limit is reached. Then the boresight is stepped by a half FOV in the direction of the spacecraft orbital motion: l btnew = l btold + a 2 and b btnew = b btold , and the scan is restarted in the opposite direction (away from the Sun): l btnew = l btold and b btnew = b btold + a 2. The exposures continue until reaching the opposite boresight envelope angle, or until time for the recovery slew to be made. The consideration of the FOV and solar battery orientation and roll angle is similar to the in-scan case.

-1 T Note that RSO  EC = REC SO = R ECSO which is a consequence of the orthogonality of matrices RSO  EC and REC  SO , which, in turn, follows from the orthogonality of matrix R (A2). Using matrices REC  SO and RSO  EC , the relations between the ecliptic and spacecraft orbit coordinates can be written in a matrix-vector form as

é cos b cos l ù é cos b cos l ù ê ú ê ú=R EC  SO ê cos b sin l ú , ê cos b sin l ú ê ú êë sin b úû êë sin b úû é cos b cos l ù é cos b cos l ù ê ú ê cos b sin l ú = RSO  EC êê cos b sin l úú. ê ú êë sin b úû êë sin b úû

(B4)

Explicit expressions for the coordinate transformations can be derived by performing the matrix-vector multiplications in Equations (B4) and rearranging the terms. The resulting formulas are: Ecliptic to spacecraft orbit coordinate transformation: l = arctan2[sin i sin b + cos i cos b ´ sin ( l - l W ) , cos b cos ( l - l W )ùû

(B5a)

b = arcsin[cos i sin b - sin i cos b sin ( l - l W )ùû .

(B5b)

C.2 Anti-Sun Observing Anti-Sun observing refers to observations along a line defined by the Earth–Sun plane that goes from the ecliptic north pole through the ecliptic south pole and passes through the anti-Sun point. Anti-Sun observing is needed for a variety of reasons, including improved photometric and astrometric coverage of binary stars and to provide additional positional information to break parallax and proper motion degeneracy. During anti-Sun observing the angle between the instrument boresight and direction to the Sun is within 180° ± 40°. In this attitude, the spacecraft solar panels are not fully illuminated, thus limiting the allowable duration of such observations to less than 1 hour per day. A sequence of astrometric exposures is carried out using the nominal 1/2 FOV advances. During the anti-Sun mode, large slews are not expected until another observing mode is desired. For modeling of anti-Sun observations, the ecliptic coordinates are most suitable. Thus, the instrument boresight longitude l bt can be expressed through the Sunʼs latitude: l bt = l ☉  180 (“+” is used when l ☉ < 180, and “−” if l ☉ ⩾ 180). The boresight latitude advancement is b btnew = b btold + s · a 2, where s = 1 or −1, and the boresight is always within the “anti-Sun” envelope: as, max ∣ b bt ∣ ⩽ qenv = 40. The FOV is aligned with the ecliptic longitude and latitude isolines: in the left part of Figure 4, the spacecraft orbit isolines around the boresight direction (l = l bt , b = b bt ) should be replaced with the respective ecliptic isolines (l = l bt ,

Spacecraft orbit to ecliptic coordinate transformation: l = arctan2 éë cos b ( cos l sin l W + sin l cos l W cos i ) - sin b cos l W sin i , cos b ( cos l cos l W - sin l sin l W cos i ) + sin b sin l W sin i]

(B6a)

b = arcsin(cos i sin b + sin i cos b sin l).

(B6b)

Note that we utilize the two-argument arc-tangent function arctan2(y,x) defined as: arctan2(y , x ) if ïìï arctan(y x ) ïï arctan(y x ) + p if ïï ï arctan(y x ) - p if = ïí if ïï + p 2 ïï - p 2 if ïï undefined if ïîï

x x x x x x

> < < = = =

0 0, 0, 0, 0, 0,

y y y y y

⩾0 0 90 . (b) Transform the ecliptic coordinates of the cell polygon vertices to the local Cartesian coordinates with the use of Algorithm 2 given below. (c) Check in the local Cartesian coordinates if a projection of the n p-vertices cell polygon on the local

where Dl and Db are small increments over λ and β, respectively: ∣ Dl ∣  a , ∣ Db ∣  a . Once Dl and Db are known, the tangent of the roll angle can found as tan q roll =

Dl . Db

(D3)

To compute Dl and Db , we expand functions Λ and B in Equations (D2a) and (D2b) in a Taylor series around the boresight point l bt, b bt and keep the constant and linear terms only (because of the smallness of δ): Dl = L ( lbt , b bt + d , i , l W ) - L ( lbt , b bt , i , l W ) » d ·

¶L ( lbt , b bt , i ), (D4a) ¶b

Db = B ( lbt , b bt + d , i ) - B ( lbt , b bt , i ) » d ·

¶B ( lbt , b bt , i ), ¶b

(D4b)

where the derivatives ¶L ¶b and ¶B ¶b can be calculated analytically by using Equations (D1a) and (D1b) and (B6a) 16



x-y plane is inside the projection of the four-vertices FOV polygon on the local x-y plane. Use Algorithm 3 in order to check results (below).

z = x‴ cos b bt + z‴ sin b bt .

Algorithm 3. To determine whether a polygon P is inside a convex polygon Q. Here is a set of rules describing the algorithm.

Algorithm 2. To transform ecliptic coordinates λ, β to the local Cartesian coordinates x, y, z at the point l bt , b bt on the celestial sphere: x = cos b sin b bt cos ( l - l bt ) - sin b cos b bt y = cos b sin ( l - l bt ) z = cos b cos b bt cos ( l - l bt ) + sin b sin b bt - 1.

Rule 1. Polygon In Polygon. For each vertex of polygon P, apply Rule 2 to determine whether the vertex is located inside polygon Q. When all vertices of P are inside Q, then the whole polygon P is inside Q, otherwise it is outside. Rule 2. Point In Polygon. For each side of polygon Q, apply Rule 3 to check the position of the vertex of polygon P in respect to the side. If the P vertex is to the right in respect to every and each Q side, then the vertex is inside Q, otherwise it is outside. Rule 3. Point and Line. Let the points A (Ax, Ay) and C (Cx, Cy) lie on a straight line. Then the following function will determine where the point B (Bx, By) is located in respect to the line:

(E1a) (E1b)

(E1c)

The derivation of formulas (E1a)–(E1c) can be accomplished in four steps. 1. Transform the ecliptic coordinates to the “native” Cartesian coordinates associated with the ecliptic coordinate system, (β, λ) → (x′, y′, z′): x ¢ = cos b cos l

(E2a)

y¢ = cos b sin l

(E2b)

z¢ = sin b .

(E2c)

(

F (A , B , C ) = ( B x - A x ) B y - C y

(

(E3a)

y  = y¢ - cos b bt sin l bt

(E3b)

z  = z¢ - sin b bt .

(E3c)

(E4a)

y‴ = -x  sin l bt + y  cos l bt

(E4b)

z‴ = z  .

(E4c)

REFERENCES Dorland, B. N., & Dudik, R. P. 2009, arXiv:0907.5248 Dorland, B. N., Dudik, R. P., Dugan, Z., & Hennessy, G. S. 2009, arXiv:0904.4516 ESA. 1997, The Hipparcos and Tycho Catalogues (ESA SP-1200; Noordwijk: ESA) Goldstein, H., Poole, C. P., Jr, & Safko, J. L. 2001, Classical Mechanics (3rd ed.; Boston, MA: Addison-Wesley) Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759 Kottwitz, S. 2011, LATEX Beginnerʼs Guide (Birmingham: UK: Packt Publishing) Lacroute, P. 1982, in Proc. Int. Coll., The Scientific Aspects of the Hipparcos Space Astrometry Mission, ed. M. A. C. Perryman, & T. D. Guyenne (Noordwijk: ESA), 3 Perryman, M. A. C., Lindegren, L., Kovalevsky, J., et al. 1997, A&A, 323, L49 Perryman, M. A. C. 2005, Astrometry in the Age of the Next Generation of Large Telescopes, 338, 3 Phillips, L. 2012, Gnuplot Cookbook (Birmingham, UK: Packt Publishing) Zacharias, N., & Dorland, B. 2006, PASP, 118, 1419

4. Transform the local Cartesian coordinates x‴, y‴, z‴ to the final local Cartesian coordinates x, y, z by rotating around the y‴ axis by b bt - 90, (x‴, y‴, z‴) → (x, y, z): x = x‴ sin b bt - z‴ cos b bt

(E5a)

y = y‴

(E5b)

)

F = 0: point B lies on the straight line F < 0: point B lies to the left from the straight line F > 0: point B lies to the right from the straight line.

3. Transform the local Cartesian coordinates x″, y″, z″ to the local Cartesian coordinates x‴, y‴, z‴ by rotating around the z″ axis by l bt , (x″, y″, z″) → (x‴, y‴, z‴): x‴ = x  cos l bt + y  sin l bt

)

- By - A y ( B x - C x )

2. Transform the “native” Cartesian to local Cartesian coordinates by transferring the coordinate center to the local point l bt , b bt with no rotations, (x′, y′, z′) → (x″, y″, z″): x  = x ¢ - cos b bt cos l bt

(E5c)

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