Vol 17, No 5 (2017) / Yuan

A prediction method for ground-based stellar occultations by ellipsoidal solar system bodies and its application

A prediction method for ground-based stellar occultations by ellipsoidal solar system bodies and its application

Yuan Ye1, 2, , Fu Yan-Ning1, Cheng Zhuo1

Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China
University of Chinese Academy of Sciences, Beijing 100049, China

† Corresponding author. E-mail: yuanye@pmo.ac.cn


Abstract: Abstract

A new programmable prediction method is developed to refine the occultation band by taking into consideration the triaxiality of an occulting body, as well as two more factors, namely, the barycenter offset of an occulting planet from the relevant planetary satellite system and the gravitational deflection of light rays due to an occulting planet. Although these factors can be neglected in most cases, it is shown that there are cases when these factors can cause a variation ranging from several tens to thousands of kilometers in the boundaries of occultation bands. Knowledge of analytic geometry simplifies the process of derivation and computation. This method is applied to long-term predictions of Jovian and Saturnian events.

Keywords: methods: analytical;methods: numerical;astrometry;ephemerides;occultations;reference systems



1 Introduction

A ground-based stellar occultation by a solar system body occurs when the light rays emitted from the star ( 1 ) are blocked by the solar system body ( ) from reaching a ground-based observer ( ). There are many reasons why we predict such an event. As a stellar occultation occurs, the combined brightness of the pair, and , varies and a high time-resolution light curve can be obtained with fast photometry. This curve provides a strong positional constraint between and , which is useful in linking optical and dynamical frames (Olkin & Elliot 1994 ; Krasinsky 1997 ; Perryman 2009 ) and improving ’s orbital solution (see MPC’s Circular; “MPC” means “Minor Planet Center” here. The website address is http://www.minorplanetcenter.net/iau/mpc.html; Folkner et al. 2014 ) or ’s motion parameters. In addition, much more information, such as the multiplicity of (Herald et al. 2010 ), ring systems or companions of (Timerson et al. 2013 ; Braga-Ribas et al. 2014 ; Ortiz et al. 2015 ; Dunham et al. 2016 ), and the physical parameters of ’s surface (Ďurech et al. 2011 ; Gomes-Júnior et al. 2015 ; Dunham et al. 2016 ) or atmosphere (Elliot & Olkin 1996 ; Christou et al. 2013 ; Westfall & Sheehan 2015 ), can be obtained.

A stellar occultation can be predicted if the motion of , the ephemeris for , and the orientation and size/shape model of are available. If the geocentric parallax of , the diurnal aberration of the occultation pair, the gravitational light deflection due to and the atmospheric refraction effects are ignored, then is a point source at infinity emitting parallel light and the side face of the shadow cast by as seen on is a column. Here, the geocentric positions of and of in the ITRS are called the proper places (Kaplan 2005 ; Urban & Seidelmann 2014 ). Though this geometric model of occultation is simple and generally has a precision of order mas for asteroids (Berthier 1997 ), it has not been fully implemented so far. In particular, a spherical is assumed in the existing prediction methods. With this assumption, Taylor ( 1955 ), Wasserman et al. ( 1981 ) and Dave Herald (the author of IOTA/WinOccult software 2 ) developed some graphical prediction methods, by which the boundaries of the occultation band can be predicted and plotted like the central line and the northern and/or southern limits of the path of a solar eclipse (Urban & Seidelmann 2014 ).

In most cases, it makes little sense to take into account the non-spherical shape of . This is because the induced errors are generally one order less than the total uncertainty of the predicted positions of and . However, this is not the case when a precisely measured is occulted by one of the four massive planets, Mars or some minor planets. For example, the difference between the polar and equatorial radii of Jupiter is thousands of kilometers (hundreds of mas; see Table 2), while the positional uncertainty ( ) of the Jovian center in INPOP10a is estimated to be about 300 km ( 30 mas) for J2000.0±100 yr (Soffel & Langhans 2013 ) and the positional uncertainty ( ) of a star in the Tycho-Gaia Astrometric Solution (TGAS) is known to be in the catalog epoch (Michalik et al. 2015 ).

Model-A Model-B Orientation and Size/Shape Model Ephemeris
Name r 1 r 2 r 3 Oblateness 7 Source ID Reference Source ID
km km km m km km
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)
Planets L/G 8 L/G L/G L/G L/–
Jupiter 71492 71492 66854 1.41 IAU/IAG IAU_Jupiter Archinal et al. ( 2011 ) JPL/jup310 9 599
71492 0.06481 66858 WinOccult JPL/DE430 5
Saturn 60268 60268 54364 0.42 IAU/IAG IAU_Saturn Archinal et al. ( 2011 ) JPL/sat375 699
60268 0.09696 54364 WinOccult JPL/DE430 6
Asteroids L/G L/G L/G L/G –/–
(3) Juno 152 143 104 DAMIT 10 A102_M103 Ďurech et al. ( 2011 ) Preston 11
141 125 110 DAMIT A102_M786 Viikinkoski et al. ( 2015 ) Preston
145 WinOccult Preston

Table 2 Model Parameters of Occulting Bodies Used in Case Studies

A more favorable situation is that a TGAS star is occulted by Saturn, in which case and are all 1 mas (Soffel & Langhans ( 2013 ); Michalik et al. ( 2015 )). In fact, like the MIT group does for some planetary or natural satellitic events (Olkin & Elliot 1994 ) and Steve Preston (hereafter referred to as Preston) does for some asteroidal events 3 , there are ways to reduce total positional uncertainties (σ), during occultation events, to 10 mas.

In the present paper, we develop a programmable prediction method that can be applied to a triaxial ellipsoidal , for which the IAU approved orientation and size/shape models are used. Consistent with this, two more previously neglected factors, namely the barycenter offset of an occulting planet from the relevant planetary satellite system (Kaplan 2005 ) and the gravitational deflection of light rays due to an occulting planet, are also taken into consideration. In fact, for Jovian and Saturnian events, these two factors can cause a variation of several hundred kilometers or more in the boundaries of occultation bands. Complete solutions for the curves and circumstances are derived based on knowledge of the associated analytic geometry.

The paper is organized as follows: in Section 2, the method is described; in Section 3, the efficacy is validated with realistic applications; in Section 4, the conclusions and future prospects are given.

2 Method

All the vectors mentioned below, e.g., and in Table 1, are written as column vectors (3-by-1 matrixes) with respect to ITRS.

Symbol Meaning
the Earth
the ground-based observer
the occulted star
the occulting solar system body or the center if its position is relevant
the positional uncertainty of in the star catalog
the positional uncertainty of in the ephemeris
σ , the total positional uncertainty of the occultation pair, and
the equatorial radius and polar radius of geocentric reference ellipsoid
the radii of -centered reference ellipsoid measured along the three principal axes
the equatorial radius and polar radius of -centered reference ellipsoid
t the time in Terrestrial Time (TT), once known as the Terrestrial Dynamical Time (TDT)
the matrix of transformation from the -fixed reference system to ITRS with light-time correction, a 3-by-3 matrix
the proper place of in ITRS, a unit vector written as a column vector, namely a 3-by-1 matrix
the proper place of ’s center in ITRS, a vector written as a 3-D column vector, namely a 3-by-1 matrix
Abbreviation/Term Explanation
ITRS International Terrestrial Reference System
GRE geocentric reference ellipsoid, i.e., an ellipsoid that approximates the geoid over the entire , to which the axes of ITRS are oriented, e.g., the World Geodetic System 1984 (WGS84)
BRS -fixed reference system
BRE -centered reference ellipsoid, i.e., an ellipsoid that approximates mean sea level (or the one-bar pressure surface) over an entire Earth-like (or large gaseous) , to which the axes of BRS are oriented
occultation band the path of the shadow cast by away from and across GRE, plotted in the map like the path of a solar eclipse
curves a collection of the important loci of the bounds of the occultation which outline and delimitate the occultation band on GRE
general circumstances a collection of the important extrema for curves of the occultation on GRE and in time, which join together the various curves
local contact times instants of time when the phenomena of ingress or egress occur in time instants
uncertainty limits limits of the ranges in which the curves and circumstances could possibly fall due to the errors induced by the positions of the occultation pair, and

Table 1 Symbols and Terms/Abbreviations

In Table 1, the proper places of and of in ITRS, the equatorial radius and polar radius of GRE, the transformation matrix from BRS to ITRS with light-time correction, and the radii of BRE measured along the three principal axes, are obtained according to (a) IAU resolutions (IAU 2001 , 2008 ), (b) the IERS convention (IERS 2010 ), and (c) the 2009 IAU report (Archinal et al. 2011), DAMIT 4 (Durech et al. 2010 ) or ISAM 5 (Marciniak et al. 2012 ) To be specific, we

read the motion parameters of from high-precision star catalogs and the position of from high-precision ephemerides, in particular, the position of a planet from the relevant natural satellite ephemerides (Moyer 2005 ; Kaplan 2005 );

calculate the proper places of and of , taking into account the motion of , the motion of with light-time correction, the light deflection due to the Sun, and the annual aberration (Kaplan 2005 ; Urban & Seidelmann 2014 );

model the surface of spheroidal as GRE with WGS84, the surface of ellipsoidal as BRE with the orientation and shape/size model of and with correction of the light deflection due to (Moyer 2005 ; Kaplan 2005 ; Urban & Seidelmann 2014 ), as well as the shadow column as seen on ; and

formulate the boundaries of the occultation band, i.e., the curves and general circumstances, and the local contact times, neglecting the remaining high-order relativistic effects (Berthier 1997 ; Moyer 2005 ; Urban & Seidelmann 2014 ).

Below, key steps in the derivation will be given.

2.1 , GRE and the Horizon Containing

Here, we give the geometric relationship between and GRE in ITRS. The equation of GRE is

where

As the normal to GRE at is , an observer at on GRE seeing in the horizon also satisfies

where is a plane through the geocenter and perpendicular to the vector . If , the observer will see appearing above the horizon. According to the formula for a tangent cylindroid of a quadric, the horizon on which is seen from GRE rising or setting is the cylindroid,
where .

2.2 Shadow Model Refinement 1 via Consideration of BRE

Like Equation (1), BRE is given by,

where, in the ITRS,

According to the formula for a tangent cylindroid of a quadric again, the shadow column of BRE without correction of light deflection of due to is the cylindroid parallel to the vector ,

where . The light deflection due to will be considered in the next subsection (Sect. 2.3).

Applying the tangent and cotangent half-angle formulas, the shadow cylindroid is parameterized by two variables, ζ and β, and

where , a 1 and a 2 are the semi-major and semi-minor axes of the perpendicular elliptic cross section of the shadow cylindroid along the directions, and , respectively, and β is the eccentric anomaly of this elliptic cross section. According to the invariant theory of a quadric, given the two positive eigenvalues of the matrix , , then , and are the unit eigenvectors corresponding to the two positive eigenvalues , respectively. The shadow axis is the main principal axis of this shadow cylindroid, namely, , where .

Consider the one-parameter (say, time t) family of shadow cylindroids , with time-dependent and . According to surface theory in classic differential geometry, the envelope of this family of cylindroids is

2.3 Shadow Model Refinement 2 via Correction of Light Deflection Due to

Consider an ellipse, for which the semi-major axis and semi-minor axis are a 1 and a 2 respectively. The eccentricity of this ellipse is . The polar form of the equation for this ellipse with its origin at the center and angular coordinate θ measured from the major axis is

It can be easily proved that

where γ and . That is to say, increasing/decreasing the same small offsets, , at points on the ellipse along their radii are almost equivalent to increasing/decreasing the same offsets along the semi-axes.

Below, we explain why the side face of planet ’s shadow in the vicinity of can still be considered as a cylindroid. The perpendicular offset from the light path grazing and deflected by massive to its asymptotic line is no greater than (Moyer 2005 ), where GM is ’s gravitational constant and c is the speed of light. Even for Jupiter, this offset is meter-scale. So, we replace the light path grazing with its asymptotic line, for which the bending angle is . The perpendicular section becomes smaller and smaller as one goes from to . For planets, it can be easily proved that

where and , and are the semi-major and semi-minor axes at respectively, and d is the distance from to . That is to say, in the vicinity of , this accumulated effect can be modeled by decreasing each semi-major and semi-minor axis of the elliptical perpendicular section of the shadow cylindroid at by factors and respectively.

As the two semi-axes of the perpendicular cross section of the shadow cylindroid along the direction are updated into , respectively, the parametric equation of the shadow cylindroid, Equation (6), is updated into

where . According to the invariant theory of a quadric again, the equation of the shadow cylindroid, Equation (5), is updated to
where

If

and

2.4 Curves and Circumstances

Below we will derive the curves in ITRS, the general circumstances and the local contact times based on their respective definitions. Also, we will model the uncertainty limits ( ) of the curves by applying WinOccult, with general circumstances and local contact times.

2.4.1 Rising and Setting Curves and Related General Circumstances

Rising and setting curves are the loci of the points on GRE at which observers see the phenomena of ingress or egress on the horizon. These points satisfy

The beginning and ending points and time instants of these loci, , are the related general circumstances at which the earliest ingress and latest egress are observed respectively. If t equals t 1 or t 2, ’s shadow cylindroid firstly or lastly contacts GRE respectively, namely,

where
and

2.4.2 Central Line and Related General Circumstances

A central line is the locus of points on GRE at which observers see the phenomenon of central occultation (possible central flash 6 ) above the horizon. These points satisfy

The beginning and ending points, and time instants of these loci, , are the related general circumstances at which observers see the phenomena of central occultation on the horizon. If t equals t 3 or t 4, ’s shadow axis firstly or lastly contacts GRE respectively, namely .

2.4.3 Northern and/or Southern Limits and Related General Circumstances

The northern (southern) limit is the locus of points on GRE at which observers see the phenomenon of grazing above the horizon. These points satisfy

The beginning and ending points, and time instants of these loci, ( ), are the related general circumstances at which observers see the phenomenon of grazing on the horizon. If t equals t 5 (t 7) or t 6 (t 8), the northern (southern) limit firstly or lastly intersects the rising and setting curves respectively, namely, . As usual, in order to display such intersections, auxiliary curves satisfying

(or equivalently ) are given. Like the case of ’s shadow cylindroid, say Equation (6), the equation describing the horizon containing , , can also be parameterized.

2.4.4 Local Contact Times

As time elapses, it is possible for an observer at a given point to see the phenomena of ingress and egress at instants of time and , respectively. It means that such a point, at the local contact times , is located on ’s shadow cylindroid and satisfies Equation (12).

2.5 Computing Methods

Equation (15) can be simplified into a constrained quadratic equation for ζ. By substituting Equation (11) for , Equation (13) can be simplified into quartic equations for x, and Equation (16) can be simplified into a constrained, real high-degree polynomial equation for x. The real roots of the quartic equation for x deduced from

can be used as the initial guesses of x for Equation (16). One choice is to solve the eigenvalue problems by the QR method and the real polynomial equations by the eigenvalue method (e.g. Press et al. 2007 ).

Solutions above give the expected locations of the curves and the expected values of the general circumstances ( ) and the local contact times ( ). Given a total positional uncertainty σ, the uncertainty limits ( ) of the curves and circumstances, as defined in WinOccult, can be solved by submitting for γ in Equation (12). We call the cylindroid, the side face of which is defined by Equation (12) with , the “outer/inner shadow,” and call the curves and circumstances of the outer/inner shadow the corresponding outer/inner “ uncertainty limits.” The uncertainty limits of the general circumstances and of the local contact times are written as and , respectively.

3 Validation and Results

The efficacy of the model derived in Section 2 is validated by comparing the following two models: Model-A, the model derived in Section 2, and Model-B, the model used in WinOccult. On one hand, the predictions using Model-B are compared with those obtained from WinOccult or from Preston’s predictions using WinOccult in order to ensure that the model is properly implemented with our code. On the other hand, the predictions using Model-A are compared with those using Model-B in order to show improvement brought by taking the ellipsoidal shape of , the barycenter offset of and the light deflection due to into consideration.

3.1 Comparison of Local Contact Times

In Table 3, the locations of the observatories witnessing the Jovian event on 2009-08-03, as well as the reported half-light times (Christou et al. 2013 ), and local contact times predicted with WinOccult, Model-B and Model-A, are listed. In this subsection, these four kinds of times are compared.

Location Christou et al. ( 2013 ) WinOccult4.2 Model-B Model-A 12
Rec Obs Lon Lat Height 1/2-light 1/2-light
# ID °E °N meter hr:mn:sc.f hr:mn:sc.f hr:mn.f hr:mn.f hr:mn.ff hr:mn.ff hr:mn.ff hr:mn.ff hr:mn.ff hr:mn.ff
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
1 TSATIG 23.893 37.998 200 22:58:14.9 24:47:22.7 22:58.7 24:47.2 22:58.73 24:47.20 22:58.38 22:58.62 24:46.94 24:47.18
2 LIAKOS 23.780 37.969 250 22:58:25.4 —:—:—.– 22:58.7 24:47.2 22:58.73 24:47.21 22:58.39 22:58.63 24:46.95 24:47.19
3 EPPICH 16.350 –23.400 1800 23:07:59.6 24:43:11.3 23:08.4 24:42.9 23:08.50 24:42.88 23:08.13 23:08.41 24:42.61 24:42.89
4 BATH 16.350 –23.400 1800 23:08:00.1 24:43:08.5 23:08.4 24:42.9 23:08.50 24:42.88 23:08.13 23:08.41 24:42.61 24:42.89
5 RCASAS 2.090 41.550 230 22:59:12.0 —:—:—.– 22:59.6 24:50.0 22:59.65 24:49.99 22:59.31 22:59.54 24:49.74 24:49.97
6 EBACAS –2.546 37.224 2170 22:59:57.7 24:50:33.1 23:00.4 24:50.4 23:00.43 24:50.36 23:00.09 23:00.32 24:50.11 24:50.35
7 SCHMAS –16.511 28.301 2390 23:02:05.9 24:51:58.3 23:02.5 24:51.8 23:02.52 24:51.77 23:02.18 23:02.42 24:51.51 24:51.75
8 MCDIAZ –16.511 28.300 2390 23:01:52.9 24:52:04.4 23:02.5 24:51.8 23:02.52 24:51.77 23:02.18 23:02.42 24:51.51 24:51.75
9 ASRIB –45.583 –22.534 1800 —:—:—.– 24:51:24.6 23:12.4 24:51.2 23:12.41 24:51.15 23:12.04 23:12.31 24:50.88 24:51.15

Notes: The numeric format of the time instants. hr: hour, mn: minute, f: fraction of minute. The lengths of them mean the numeric lengths.

Table 3 Local Contact Times in UTC of the Jovian Event on 2009–08–03

Firstly, the expected ingress and egress times predicted with Model-B are compared with those predicted with WinOccult. In both predictions, the UCAC4 position of and the JPL/DE430 ephemeris for are used. The differences between the two kinds of times are almost the same, indicating that our code using Model-B gives the same predictions as WinOccult. Therefore, we will only focus on comparisons between Model-B and Model-A.

Second, the expected ingress and egress times predicted with Model-A are compared with those predicted with Model-B. In predictions using Model-A, the relevant natural satellite ephemeris used is the JPL/jup310 ephemeris. As can be seen from Table 3, the mean difference between the predicted egress times is 0.26 min, comparable to the 1σ error introduced by the uncertainties in positions of and . Since almost the same oblate shape of Jupiter is used in both models, the difference is caused by the barycenter offset of and the light deflection due to .

3.2 Comparison of General Contact Times

In Table 4, the general contact times in TDT of the Jupiter and Saturn events from 2017 to 2036 are listed, together with some other helpful information. A subscript “A” or “B” is added if Model-A or Model-B is used for prediction respectively. Like WinOccult, the predictions are limited to solar elongations of no less than 12 degrees and to visual magnitudes of stellar candidates no fainter than 9.0 mag for Jovian events and 10.5 mag for Saturnian events by default, and the positional uncertainty of the JPL ephemerides is assumed to be 50 mas. Here, the stellar candidates are chosen from the star catalogs UCAC4 (Zacharias et al. 2013 ) and URAT1 (Zacharias et al. 2015 ). In this subsection, the general contact times predicted with Model-A are compared with those predicted with Model-B. The comparisons are shown in Figure 1.

Fig. 1 Top left: Histogram of the differences between the expected values of Model-A and Model-B . Top right: Histogram of the 3σ errors of Model-A . Bottom: Histogram of the ratios of the differences and the errors of Model-A plotted in the top panels. The solid lines represent t 1 and the dashed lines lines represent t 2. Data are listed in Table 4.
Rec Model-A Model-B Occultation pair Positional Data Size/Shape
# Date Source σ Model
yyyy-mm-dd hr:mn:sc hr:mn:sc hr:mn:sc hr:mn:sc hr:mn:sc hr:mn:sc ID Name mas km ID
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
1 2018-07-29 10:22:26 10:23:50 15:06:43 15:08:05 10:22:04 15:07:08 HIP 72182 Jupiter UCAC4+JPL 60 225 IAU_Jupiter
2 2018-08-08 02:49:47 02:50:50 05:35:03 05:36:05 02:48:32 05:37:10 HIP 72417 Jupiter UCAC4+JPL 58 223 IAU_Jupiter
3 2019-10-21 11:06:15 11:06:43 12:30:54 12:31:23 11:06:47 12:31:02 HIP 84936 Jupiter UCAC4+JPL 70 294 IAU_Jupiter
4 2020-12-21 06:18:07 06:18:29 07:14:13 07:14:36 06:17:51 07:14:55 HIP 99314 Jupiter UCAC4+JPL 58 249 IAU_Jupiter
5 2021-01-12 16:56:05 16:56:25 17:52:50 17:53:11 16:56:13 17:53:12 TYC 6337-01274-1 Jupiter UCAC4+JPL 62 272 IAU_Jupiter
6 2021-04-02 08:45:16 08:45:38 10:03:19 10:03:41 08:45:31 10:03:22 HIP 107232 Jupiter UCAC4+JPL 58 238 IAU_Jupiter
7 2024-04-03 21:11:39 21:12:09 21:54:18 21:54:48 21:10:17 21:56:16 HIP 14089 Jupiter UCAC4+JPL 55 232 IAU_Jupiter
8 2025-04-26 23:06:56 23:07:36 24:18:28 24:19:08 23:07:19 24:18:47 HIP 24668B Jupiter UCAC4+JPL 103 434 IAU_Jupiter
9 2025-04-26 23:07:00 23:07:22 24:21:08 24:21:30 23:07:14 24:21:16 HIP 24668A Jupiter UCAC4+JPL 57 240 IAU_Jupiter
10 2025-10-13 12:28:42 12:29:31 15:34:47 15:35:36 12:29:17 15:34:46 HIP 37442 Jupiter UCAC4+JPL 61 228 IAU_Jupiter
11 2029-11-28 13:33:49 13:34:13 14:36:28 14:36:52 13:34:01 14:36:35 HIP 72182 Jupiter UCAC4+JPL 68 312 IAU_Jupiter
12 2030-08-27 09:18:55 09:20:09 10:28:31 10:29:45 09:15:02 10:33:29 HIP 74299 Jupiter UCAC4+JPL 61 244 IAU_Jupiter
13 2031-05-21 21:46:43 21:47:45 24:28:59 24:30:00 21:45:48 24:31:15 TYC 6828-00196-1 Jupiter UCAC4+JPL 71 225 IAU_Jupiter
14 2031-11-29 13:24:44 13:25:09 14:29:27 14:29:53 13:25:09 14:29:33 TYC 6843-01786-1 Jupiter UCAC4+JPL 77 340 IAU_Jupiter
15 2031-12-08 15:55:01 15:55:54 16:32:58 16:33:51 15:53:51 16:34:43 TYC 6844-00458-1 Jupiter UCAC4+JPL 96 428 IAU_Jupiter
16 2032-04-23 13:09:39 13:10:35 16:47:16 16:48:13 13:10:23 16:47:18 HIP 99501 Jupiter UCAC4+JPL 62 226 IAU_Jupiter
17 2035-06-28 22:10:01 22:10:33 23:11:56 23:12:28 22:09:05 23:13:38 HIP 12616 Jupiter UCAC4+JPL 59 235 IAU_Jupiter
18 2036-03-15 17:11:53 17:12:19 18:25:21 18:25:48 17:11:40 18:25:59 HIP 14089 Jupiter UCAC4+JPL 59 238 IAU_Jupiter
19 2018-07-05 05:04:40 05:06:02 06:29:46 06:31:11 05:04:40 06:32:42 TYC 6277-00323-1 Saturn UCAC4+JPL 62 407 IAU_Saturn
20 2018-07-12 02:16:20 02:17:50 03:59:45 04:01:17 02:17:17 04:01:30 TYC 6844-02804-1 Saturn UCAC4+JPL 77 507 IAU_Saturn
21 2021-06-28 24:14:46 24:16:18 02:31:34 02:33:01 24:11:35 02:31:14 TYC 6349-00492-1 Saturn UCAC4+JPL 58 383 IAU_Saturn
22 2022-01-07 13:12:59 13:16:03 13:26:21 13:29:20 13:07:18 13:31:43 TYC 6349-00662-1 Saturn UCAC4+JPL 59 462 IAU_Saturn
23 2023-04-15 05:43:35 05:44:28 07:24:45 07:01:38 05:43:41 07:01:27 TYC 5807-01344-1 Saturn UCAC4+JPL 66 498 IAU_Saturn
24 2024-08-06 16:27:26 16:30:48 17:47:12 17:50:50 16:28:36 17:58:52 1UT 417-493954 Saturn URAT1+JPL 89 569 IAU_Saturn
25 2026-11-17 09:11:20 09:14:33 11:24:32 11:27:33 09:02:00 11:27:15 TYC 10-00114-1 Saturn UCAC4+JPL 66 417 IAU_Saturn
26 2030-03-07 21:56:27 21:57:32 23:21:19 23:22:26 21:57:30 23:23:22 TYC 1225-00469-1 Saturn UCAC4+JPL 65 449 IAU_Saturn
27 2030-04-06 21:54:10 21:54:54 22:55:56 22:56:40 21:54:48 22:56:46 TYC 1233-00635-1 Saturn UCAC4+JPL 64 459 IAU_Saturn
28 2030-04-23 09:05:10 09:07:17 09:32:57 09:35:08 09:04:49 09:37:11 TYC 1238-00068-1 Saturn UCAC4+JPL 97 705 IAU_Saturn
29 2030-04-25 17:18:51 17:19:34 18:10:13 18:10:58 17:19:27 18:11:27 TYC 1238-00025-1 Saturn UCAC4+JPL 65 473 IAU_Saturn
30 2030-08-03 06:27:10 06:28:35 08:06:07 08:07:31 06:28:38 08:06:19 TYC 1272-00299-1 Saturn UCAC4+JPL 93 640 IAU_Saturn
31 2030-08-17 20:36:39 20:40:02 21:34:28 21:37:49 20:31:05 21:42:43 TYC 1273-00453-1 Saturn UCAC4+JPL 65 436 IAU_Saturn
I 2031-06-26 —:—:— —:—:— 02:03:03 02:23:50 TYC 1292-01822-1 Saturn UCAC4+JPL 60 435 IAU_Saturn
32 2032-04-07 05:33:07 05:34:02 07:04:51 07:05:46 05:33:46 07:04:40 HIP 23883 Saturn UCAC4+JPL 65 447 IAU_Saturn
33 2032-04-15 03:04:17 03:04:59 04:24:59 04:25:40 03:05:01 04:25:12 HIP 24129 Saturn UCAC4+JPL 54 375 IAU_Saturn
34 2032-08-17 22:14:18 22:15:25 23:08:21 23:09:27 22:12:48 23:09:55 TYC 1326-01020-1 Saturn UCAC4+JPL 58 404 IAU_Saturn
35 2033-01-14 22:11:14 22:12:19 24:12:55 24:14:01 22:12:05 24:13:37 TYC 1864-01637-1 Saturn UCAC4+JPL 63 370 IAU_Saturn
36 2033-04-20 01:48:12 01:49:06 03:23:01 03:23:55 01:49:09 03:23:11 TYC 1864-01336-1 Saturn UCAC4+JPL 62 425 IAU_Saturn
37 2034-08-01 08:43:05 08:43:40 09:39:42 09:40:17 08:43:04 09:39:40 TYC 1375-01469-1 Saturn UCAC4+JPL 61 444 IAU_Saturn
38 2034-08-06 08:23:14 08:23:52 09:11:34 09:12:10 08:22:15 09:11:45 HIP 38777 Saturn UCAC4+JPL 53 385 IAU_Saturn
39 2035-01-10 03:21:47 03:22:40 05:14:53 05:15:47 03:22:34 05:16:14 TYC 1386-00517-1 Saturn UCAC4+JPL 58 341 IAU_Saturn
40 2036-03-07 15:44:53 15:46:19 18:01:44 18:03:14 15:45:54 18:04:55 TYC 1397-00363-1 Saturn UCAC4+JPL 67 404 IAU_Saturn
41 2036-07-26 10:34:42 10:35:56 11:05:48 11:07:08 10:34:52 11:10:11 TYC 1403-00072-1 Saturn UCAC4+JPL 69 508 IAU_Saturn
42 2036-07-29 12:08:17 12:11:00 12:22:26 12:25:14 12:06:58 12:30:19 HIP 46745 Saturn UCAC4+JPL 68 501 IAU_Saturn
II 2015-04-01 01:06:07 01:06:11 01:27:59 01:28:03 01:06:11 01:28:00 TYC 0803-00156-1 (3) Juno TGAS+Preston 8 11 A102_M103
2015-04-01 01:06:08 01:06:12 01:27:58 01:28:02 01:06:12 01:27:58 TYC 0803-00156-1 (3) Juno TGAS+Preston 8 11 A102_M786

Table 4 General Contact Times in TDT and Information on Sample Predictions with Model-A/B

The top-left panel of Figure 1 shows that the differences between the general contact times predicted with Model-A and Model-B can reach about 10 minutes. Because of the rotation of Earth, such differences can cause variations in length of bands ranging from tens to hundreds of kilometers. In order to show that the above significant improvement gained with Model A is meaningful, in the top-right panel of Figure 1 we plot the histogram of the 3σ errors induced by uncertainties in the positions of and . The bottom panel of Figure 1 tells us that the gained improvement can be 5 times greater than the 3σ errors, and the improvements in 50% of cases exceed the 3σ errors.

Although the barycenter offset of and the light deflection due to do influence the predictions, it can be estimated that the total variations in the proper places of and caused by these two factors (Kaplan 2005 ) are always much smaller than 3σ values ( mas) in Table 4. Therefore, the remaining factor, the non-spherical shape of , causes variations in predictions exceeding the 3σ errors.

3.3 Comparison of Locations and Widths of Bands

3.3.1 The (3) Juno Event

In this subsection, the locations and widths of the bands for Preston’s prediction for the (3) Juno events 13 and those predicted with our code using Model-B and Model-A are compared.

Firstly, curves predicted with Model-B in the left panel of Figure 2 are compared with Preston’s prediction. In both predictions, the URAT1 position of and Preston’s orbital solution for are used. The uncertainty of the URAT1 position is 10 mas and the uncertainty of Preston’s orbital solution is merely 8 mas. The two kinds of curves are almost the same, indicating that our code using Model-B gives the same predictions as WinOccult. Therefore, we will only focus on comparisons between Model-B and Model-A.

Fig. 2 Left: Map of the Model-B curves and 1, 2 and 3σ limits for the (3) Juno events based on the URAT1 catalog (used by Preston) and Preston’s pre-event orbital solution, where σ is the total positional uncertainty, 13 mas. Right: Map of the Model-B curves and 1, 2 and 3σ limits for the (3) Juno events based on the TGAS catalog and Preston’s pre-event orbital solution, where σ is merely the positional uncertainty of the ephemeris, 8 mas. The dashed lines represent the 1, 2 and 3σ limits of curves. The red crosses represent observatories with negative observations and green points represent the observatories with successful observations. The record numbers are listed in Table 4.

Second, the improvement in the location of the band is considered. Since the orbital solution of is already determined and updated by Preston with last-minute astrometry, we merely update the position of . As TGAS is the most high-precision star catalog currently available (Michalik et al. 2015 ) and is independent of this stellar occultation, the position of is updated to the TGAS position, for which the uncertainty is less than 1 mas and so neglected. As shown in the right panel of Figure 2, compared with previous predictions, the northern limit moves closer to the “negative” station. That is to say, the predictions for local visibilities are improved via refinement in the prediction for the central line. However, the misprediction of the “negative” station is still made with Model-B.

Third, variations in predictions caused by the ellipsoidal shape of are considered. Given the two kinds of DAMIT models of (3) Juno, A102_M103 and A102_M786 in Table 2, we again predict the bands for this event with Model-A. The TGAS position and Preston’s orbital solution are still applied. Compared with the previous prediction using Model-B in the left panel of Figure 2, the bands predicted with Model-A in Figure 3 become narrow. No matter which kind of DAMIT model is applied, the “negative” station is located outside the band. To conclude, Model-A is helpful in predictions of local visibilities.

Fig. 3 The left column uses A102_M103 and the right column uses A102_M786 as the size/shape model of (3) Juno. Top row: Sky-projections of (3) Juno at the middle time of the (3) Juno event listed in Table 4. They are available from the DAMIT website. Bottom row: Maps of the Model-A curves and 1, 2 and 3σ limits for the (3) Juno events based on TGAS catalog and Preston’s pre-event orbital solution, where σ is merely the positional uncertainty of the ephemeris, 8 mas. The dashed lines represent the 1, 2 and 3σ limits of curves. The red crosses represent observatories with negative observations and the green points represent observatories with successful observations.

3.3.2 The Jovian and Saturnian Events

In Figure 4, the bands of Jovian and Saturnian events listed in Table 4 are plotted by using Model-A. The 3σ errors of the locations of bands are indicated by the 3σ limits of the curves. As discussed in Section 3.2, only non-spherical shapes of s cause variations in curves exceeding the 3σ errors. In order to show significant improvements in the width of bands caused by the ellipsoidal shapes of s, the locations of the outer 3σ limits of the curves for grazing events are compared with the expected curves predicted with Model-B. The curves predicted with Model-B are plotted in Figure 5.

Fig. 4 Maps of the Model-A curves and 3σ limits of the sample Jovian/Saturnian events listed in Table 4. The red lines represent the rising and setting curves, the blue lines represent the central line and the green lines represent the northern or southern limits. The 3σ limits of curves (dashed lines) are drawn, as well as sub-stellar points (red diamonds), sub-solar points (green diamonds) and sub-lunar points (blue diamonds). The record numbers are listed in Table 4, as well as some helpful information.
Fig. 5 First seven: Maps of Model-B curves and 3σ limits of the sample Jovian/Saturnian events listed in Table 4 for which northern or southern Model-A limits exist (see Fig. 4). Eighth (#I-B): Map of Model-B curves and 3σ limits of the Saturnian event for which Model-A curves do not exist. Ninth (#I-A): Map of the Saturnian events for which Model-A curves do not exist. The red lines represent the rising and setting curves, the blue lines represent the central line and the green lines represent the northern or southern limits. The 3σ limits of curves (dashed lines) are drawn, as well as sub-stellar points (red diamonds), sub-solar points (green diamonds) and sub-lunar points (blue diamonds). The record numbers are listed in Table 4, as well as some helpful information.

In comparisons between Figures 4 and 5, the expected curves predicted with Model-B are located outside the 3σ limit of the curves predicted with Model-A. In particular, as shown in the eighth and ninth panels of Figure 5, the expected curves vanish when the prediction model changes from Model-B to Model-A. That is to say, for the Jovian and Saturnian events, the variations in the width of bands due to the ellipsoidal shape of can be so large that not only the local visibilities but also the general visibilities are influenced.

3.4 Application: Long-term Predictions of Jovian and Saturnian Events

As is mentioned in Section 3.3.2, 43 Jovian and Saturnian events from 2017 to 2036 are predicted in Table 4, Figure 4 and the ninth panel of Figure 5. They represent good opportunities to directly tie the optical reference frames to the dynamical reference frames with a precision on the order of 1 mas (Olkin & Elliot 1994 ), and to study the Jovian/Saturnian atmosphere with kilometric accuracy (Elliot & Olkin 1996 ). Christou et al ( 2013 ) discussed the scientific values of the Jovian event on 2021 April 2 in terms of the Jovian atmosphere.

Among these 43 events, there are some special events, such as the eight possible grazing events (see record numbers in Fig. 5) and the seven possible central events (possible cental flashes). Grazing events probe the higher and thinner atmospheres more easily than other cases, as the drop in signals from due to high atmospheres in lasts longer for grazing events than for other cases. Central flashes probe the lower and thicker atmospheres more easily than other cases, as signals from increase due to the focusing of rays by the limbs of thick atmospheres in . Also, information on the shapes of the atmospheres can be provided by detailed structures of central flashes.

4 Conclusions

According to the results from our test cases, improvements gained by using Model-A (i.e., our model derived in Sect. 2) can exceed the 3σ errors induced by uncertainties in the positions of and . Not only local but also general visibilities are influenced by variations in the predictions caused by the ellipsoidal shape of , the barycenter offset of and light deflection due to .

The first Gaia data release (Gaia-DR1) and its subset catalog called the TGAS, which became available on line on 2016 September 14 14 , is useful for a large number of occultation predictions. Gaia-DR1 includes improved astrometric positions for about one billion stars and TGAS provides improved proper motion information for about two million Tycho stars. Dave Herald also provided a Gaia-based catalog to 14 mag used for predictions of occultations by asteroids, but we will have to wait another few data releases for all astrometry of solar system bodies. This means that, for a stellar occultation, the greatest uncertainty in the location of (the central line of) the band merely comes from positional uncertainty in the pre-event orbital solution for . Please see the presentation 15 on the website of the IOTA meeting for further details on Gaia’s impacts on occultation predictions.

Anyway, orbital solutions from can be gradually improved if observations are individually available, and there will be more and more ellipsoidal solar system bodies for which 1σ (or even 3σ) uncertainties of the orbital solutions are less than the errors caused by non-spherical shapes.


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Cite this article: Yuan Ye, Fu Yan-Ning, Cheng Zhuo. A prediction method for ground-based stellar occultations by ellipsoidal solar system bodies and its application. Res. Astron. Astrophys. 2017; 5:045.

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