Mission Planning Approach for an Exoplanet

Characterization Satellite

Javier Fernández-Villacañas Cabezas

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor: Prof. Paulo Jorge Soares Gil Co-supervisor: Eng. Antonio Gutiérrez Peña

Examination Committee

Chairperson: Prof. Fernando José Parracho Lau Supervisor: Prof. Paulo Jorge Soares Gil

Members of the Committee: Prof. Agostinho Rui Alves da Fonseca

September 2014

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Acknowledgements

To my supervisors Paulo Gil from Instituto Superior Técnico and Antonio Gutiérrez from

DEIMOS Engenharia. I thank them for all the knowledge and experience they have transmitted

to me.

To my family, who have been always there, appreciating my work, supporting my

decisions and helping me when I needed it. I particularly acknowledge my parents and my

grandma for the education and support they have given to me, and to my grandparents that will

be proudly observing this. Grandpa Nicolás, I know you would be very proud of this, since I was

a kid you knew I was going to become an engineer.

To my friends from all around the world that have contributed in my personal

development. I want to especially thank Matilde, Arantxa, Miriam and Luis, you are a part of

who I am now. Thank you very much for all these years of friendship, and for being with me in

the bad and good moments.

Last but not least, to my colleagues from DEIMOS Engenharia who accepted me as one

of their own from the beginning.

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Abstract

As the space observation techniques evolve, more astronomical objects and events are

discovered. This large number of observable objects need to be precisely characterized by

observation satellites, therefore a good planning of the observations is required. In addition,

observation satellites are orbiting a planet, typically the Earth. Consequently, the target visibility

will be affected by several constraints that will produce interruptions in the observation because

of either the occultation of the targets or the possibility of damage in the sensor.

In this thesis we will optimize the observations of an exoplanet characterization satellite,

considering several parameters. In order to proceed with the optimization, we need to calculate

the visibility of the targets by determining its constraints, like the occultation produced when the

Earth is between the satellite and the target.

Keywords: visibility of the targets, exoplanet characterization satellite, planning of the observations, observations optimization, visibility and optimization constraints.

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Resumo

A medida que as técnicas de observação evoluem, mais eventos astronómicos e

corpos celestes são descobertos. Este grande número de objetos observáveis precisa de ser

cerrectament caracterizado mediante satélites de observação, portanto é requerida uma boa

planificação das observações. Além disso, os satélites de observação estão a orbitar um

planeta, tipicamente a Terra. É por isso que a visibilidade dos objetivos vai estar afetada por

varias restrições, que vão produzir interrupções nas observações causadas pela ocultação dos

objetivos ou pela possibilidade de avariar o sensor.

Nesta tese vamos optimizar as observações de um satélite de caracterização de exoplanetas,

considerando vários parâmetros. Com o fim de proceder à optimização, temos que calcular a

visibilidade dos alvos mediante a determinação das restrições de visibilidade, como a ocultação

produzida quando a Terra está entre o satélite e o alvo.

Palavras chave: visibilidade dos alvos, satélite de caracterização de exoplanetas, planeamento das observações, optimização das observações, restrições de visibilidade e

optimização.

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Contents List of Figures _______________________________________________________________________ vii

List of Tables ________________________________________________________________________ ix

List of Acronyms ______________________________________________________________________ x

List of Symbols ______________________________________________________________________ xi

Chapter 1: Introduction ________________________________________________________________ 1

1.1 Thesis Objective _________________________________________________________________ 1

1.2 Observing the sky from orbit ______________________________________________________ 1 1.2.1 Exoplanets __________________________________________________________________ 1 1.2.2 Planetary transits and observation methods _______________________________________ 2 1.2.3 Relevant missions ____________________________________________________________ 2

1.3 Sky visibility and planning problem _________________________________________________ 4 1.3.1 Introduction to the problem ____________________________________________________ 4 1.3.2 Previous solutions ____________________________________________________________ 4 1.3.3 CHEOPS mission summary _____________________________________________________ 5

1.4 Goals of the thesis _______________________________________________________________ 8

Chapter 2: Visibility computation ________________________________________________________ 9

2.1 Definition of targets _____________________________________________________________ 9

2.2 Orbit propagation _______________________________________________________________ 9

2.3 Constraints and visibility calculation _______________________________________________ 15 2.3.1 Targets transit visibility _______________________________________________________ 15 2.3.2 Constraint of the occultation by the Earth ________________________________________ 19 2.3.3 Constraint of the Moon's exclusion angle ________________________________________ 25 2.3.4 Constraint of the Sun's exclusion angle __________________________________________ 30 2.3.5 The Earth's stray light constraint _______________________________________________ 37 2.3.6 South Atlantic Anomaly constraint ______________________________________________ 49

2.4 Combination of orbit, constraints and targets to compute visibility windows ______________ 53

2.5 Discussion of the results _________________________________________________________ 57

Chapter 3: Mission Planning ___________________________________________________________ 59

3.1 Concept of planning and scheduling ________________________________________________ 59 3.1.1 Definition _________________________________________________________________ 59 3.1.2 Class of complexity and the importance of the visibility windows _____________________ 60

3.2 Optimization of the observations __________________________________________________ 61 3.2.1 Definition of the cost function _________________________________________________ 62 3.2.2 Common Artificial Intelligence approaches _______________________________________ 62

3.2.2.1 Definition of the problem _________________________________________________ 62 3.2.2.2 Resolution methods _____________________________________________________ 64

3.2.3 Description of the algorithm employed and results ________________________________ 64

3.3 Discussion of the results _________________________________________________________ 71

Chapter 4: Conclusions and future improvements __________________________________________ 72

Bibliography ________________________________________________________________________ 74

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List of Figures Figure 2.1: Sun-synchronous orbit [14]____________________________________________________ 10 Figure 2.2: CHEOPS' orbit 3D in 𝒕𝟎 _______________________________________________________ 13 Figure 2.3: CHEOPS' orbit 2D in 𝒕𝟎 _______________________________________________________ 14 Figure 2.4: Observation interval definition _________________________________________________ 16 Figure 2.5: Nominal interval of observation for 50 targets and 1.5 months _______________________ 18 Figure 2.6: Eclipse half cone angle 𝜸𝑬𝒐𝒄𝒄 definition, not to scale _______________________________ 19 Figure 2.7: Spacecraft centered celestial sphere considering the constraint of the occultation by the Earth. Case 1. The Earth's occultation region is represented by a blue circle ______________________ 20 Figure 2.8: Spacecraft centered celestial sphere considering the constraint of the occultation by the Earth. Case 2. The Earth's occultation region is represented by a blue circle, the visible targets by green stars and the hidden targets by red squares _______________________________________________ 21 Figure 2.9: Visibility criteria for the constraint of the occultation by the Earth, comparison between 𝜽𝑬𝒐𝒄𝒄 and 𝜸𝑬𝒐𝒄𝒄, not to scale ________________________________________________________________ 22 Figure 2.10: Time intervals when the satellite is occulted by the Earth for 50 targets and 1 orbit ______ 23 Figure 2.11: Time intervals when the satellite is occulted by the Earth for 50 targets and 1.5 days ____ 23 Figure 2.12: Visibility window for the constraint of the occultation by the Earth for 50 targets and 1.5 days _______________________________________________________________________________ 25 Figure 2.13: Visibility criteria for the constraint of the occultation by the Moon, comparison between 𝜽𝑴𝒐𝒄𝒄 and 𝜸𝑴𝒐𝒄𝒄, not to scale __________________________________________________________ 26 Figure 2.14: Spacecraft centered celestial sphere considering the constraint of the Moon's exclusion angle ______________________________________________________________________________ 27 Figure 2.15: Time intervals when the satellite is affected by the Moon's exclusion region for all the targets and 6.5 days __________________________________________________________________ 28 Figure 2.16: Visibility window for the constraint of the Moon's exclusion angle for all the targets and 6.5 days _______________________________________________________________________________ 29 Figure 2.17: Visibility criteria for the constraint of the Sun's exclusion angle, comparison between 𝜽𝒂𝑺 and 𝜸𝒂𝑺, not to scale __________________________________________________________________ 30 Figure 2.18: Method of Aristarchus to measure the relative size and distance of the Sun, the Earth and the Moon, not to scale ________________________________________________________________ 31 Figure 2.19: Spacecraft centered celestial sphere considering the Sun's exclusion angle constraint ____ 32 Figure 2.20: Exclusion by the Sun time intervals for all the targets and 1.5 months _________________ 33 Figure 2.21: Diagram to explain the algorithm of the visibility time intervals considering the Sun's exclusion angle constraint: none observation can be affected by the Sun's exclusion region _________ 34 Figure 2.22: Visibility window for the constraint of the Sun's exclusion angle for all the targets and 1.5 months ____________________________________________________________________________ 36 Figure 2.23: Diagram of the intersection between the Earth and the plane 𝝅, which is defined by the satellite position and the vectors 𝒓�⃗ 𝑳𝒐𝑺 and 𝒓�⃗ 𝑬/𝒔𝒂𝒕, not to scale _________________________________ 37 Figure 2.24: Change of reference frame from 𝒙𝒚𝒛 (ECI) to 𝒙′𝒚′𝒛′ defined by the vectors 𝒓�⃗ 𝑳𝒐𝑺, 𝒖��⃗ 𝝅 and 𝒏��⃗ 𝝅, not to scale _________________________________________________________________________ 38 Figure 2.25: Upper view of plane 𝝅, definition of the tangent points T and C, not to scale ___________ 39 Figure 2.26: 3D representation of the intersection between lines 𝒔 and 𝒑 ________________________ 41 Figure 2.27: 3D representation of the case 𝑬𝑪 ≤ 𝑬𝑻, the black dot is CHEOPS' position, the orange cone is the visibility cone, the yellow and blue areas are the illuminated and dark parts of the Earth, respectively _________________________________________________________________________ 42 Figure 2.28: 3D representation of the case 𝑬𝑪 ≤ 𝑬𝑻, 1) ______________________________________ 43 Figure 2.29: 3D representation of the case 𝑬𝑪 ≤ 𝑬𝑻, 2) ______________________________________ 43 Figure 2.30: representation of the case 𝑬𝑪 ≤ 𝑬𝑻, 3) ________________________________________ 43

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Figure 2.31: Descriptive diagram of the terminator, which is the circle defined by the point E and the vectors 𝒖��⃗ , 𝒗��⃗ and 𝒓�⃗ 𝑺/𝑬, not to scale _______________________________________________________ 44 Figure 2.32: Descriptive diagram of 𝜽𝒔𝒕𝒓, which is the angle formed by 𝒓�⃗ 𝑳𝒐𝑺 and 𝒓�⃗ 𝒕𝒆𝒓𝒎/𝒔𝒂𝒕, not to scale 45 Figure 2.33: Time intervals for the Earth's stray light constraint for all the targets and 5 days 17 hours 47 Figure 2.34: Visibility window for the Earth's stray light constraint for all the targets and 5 days 17 hours __________________________________________________________________________________ 48 Figure 2.35: SAA region observed in the celestial sphere from the J2000 reference frame. Satellite outside the region __________________________________________________________________________ 49 Figure 2.36: SAA region observed in the celestial sphere from the J2000 reference frame. Satellite inside the region __________________________________________________________________________ 49 Figure 2.37: Representation of the 𝑮𝑺𝑻 angle respect to the J2000 reference frame _______________ 50 Figure 2.38: Visibility window for SAA constraint for all the targets and 4 days days 20 hours ________ 52 Figure 2.39: Spacecraft centered celestial sphere considering all the constraints. Satellite outside the SAA region _____________________________________________________________________________ 53 Figure 2.40: Spacecraft centered celestial sphere considering all the constraints. Satellite inside the SAA region _____________________________________________________________________________ 54 Figure 2.41: Effect of each constraint on the number of visible targets __________________________ 55 Figure 2.42: Visibility window considering all the constraints and targets for 3 days ________________ 56 Figure 3.1: Euler diagram for P, NP, NP-complete, and NP-hard set of problems ___________________ 60 Figure 3.2: Random nominal intervals selected considering the Sun's exclusion constraint and only the third optimization constraint for all the targets and 1.5 months _______________________________ 66 Figure 3.3: Random nominal intervals selected considering the Sun's exclusion constraint and all the optimization constraints for all the targets and 1.5 months ___________________________________ 67 Figure 3.4: Visibility random solution for all the targets and 1.5 months _________________________ 68 Figure 3.5: Zoom in Figure 3.4 to observe the time intervals when the target is not observable _______ 69 Figure 3.6: Solutions graph considering the total effective time (in blue) and the number of targets observed (in green) for 1.5 months. The red markers represent the optimal solutions ______________ 70 Figure 3.7: Calculation time considering the number of solutions computed ______________________ 70

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List of Tables Table 2.1: Classical orbital parameters for CHEOPS' orbit _____________________________________ 11 Table 2.2: Deviation angles from considering the sunlight parallel to the Earth's surface or centered on the Sun's nucleus _____________________________________________________________________ 31 Table 2.3: Sky visibility for each constraint for a general instant of time _________________________ 54 Table 3.1: Calculation time comparison for the two methods considered: 1) calculating the whole occultation region 2) comparing the actual and required half cone angles _______________________ 61

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List of Acronyms AOCS Attitude and Orbital Control Systems APE Absolute Performance Error ECI Earth-Centered Inertial ET Ephemeris Time GAP Generalized Assignment Problem GST Greenwich Sidereal Time ILP Integer Linear Programming LEO Low Earth Orbit LoS Line of sight LTAN Local Time of Ascending Node NP Nondeterministic Polynomial P Deterministic Polynomial RA Right Ascension SAA South Atlantic Anomaly TBC To Be Confirmed UTC Coordinated Universal Time

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List of Symbols Latin Symbols

𝒂 Semi-major axis 𝒂𝑺 Anti-Sun 𝒄�⃗ 𝒕 Position vector of the center of the terminator 𝑪𝒑𝒊 Coefficient of priorities 𝑪𝑹𝑭 Rotation matrix 𝒅 Sensor adjustment time 𝒆 Eccentricity 𝑬 Earth 𝑬𝒐𝒄𝒄 Constraint of the occultation by the Earth 𝑬𝑪�����⃗ Vector from the Earth's center to the intersection point between lines

𝑝 and 𝑠 𝑬𝑻�����⃗ Vector from the Earth's center to the tangent point of its surface

contained in line 𝑝 𝒇 Cost function 𝒊 Inclination 𝒊 Subscript for the target considered 𝑱𝟐 Second zonal harmonic of the Earth; 𝐽2 = 0.00108263 𝒌 Instant of time considered 𝑴 Mean anomaly and 𝑴 Subscript that means the Moon 𝑴𝒐𝒄𝒄 Constraint of the occultation by the Moon 𝒏 Mean orbital velocity 𝒏��⃗ 𝝅 Normal vector to the plane π 𝒑 Semi-latus rectum 𝑷𝒕 Period of the exoplanet revolution around the host star 𝒒𝒅𝒊𝒗 Number of orbits 𝒒𝑬 Number of revolutions of the Earth expressed as a natural number 𝒓�⃗ Position vector 𝒓�⃗ 𝑳𝒐𝑺 LoS direction vector 𝒓�⃗ 𝒔𝒂𝒕/𝑬𝑭 Position vector of the satellite in the orbit reference frame (F) 𝒓�⃗ 𝒔𝒂𝒕/𝑬𝑹 Position vector of the satellite in the J2000 reference frame 𝑹⊕ Medium radius of the Earth; 𝑅⊕ = 6378km 𝒔 Visibility time intervals considering the Sun's exclusion angle 𝑺 Sun 𝒔𝒂𝒕 Satellite 𝒔𝒕𝒓 Stray Light 𝒕 General instant of time 𝑻 Orbital period 𝒕𝒄 Instant of time when a transit occurs 𝒕𝒄𝒊/𝟎 Instant of time when the first transit occurs for the target 𝑖 𝒕𝒄𝒊/𝒏𝒑 Time instant centered on the transit when the transit takes place 𝑻𝒅𝒊𝒗 Time interval division in during one orbit 𝒕𝑬 time during each revolution of the Earth 𝒕𝒆𝒓𝒎 Terminator 𝒕𝟎 Reference time 𝒕𝒇 End of the mission instant of time 𝑻𝟎 Perigee passage time 𝒖��⃗ 𝝅 Director vector of the 𝑦′ axis 𝒖��⃗ 𝒕 Unit vector from 𝑐 towards the terminator

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𝒗 Exclusion by the Sun time intervals 𝒗��⃗ 𝒔 Director vector of line 𝑠 𝒗��⃗ 𝒕 Unit vector of the terminator perpendicular to 𝑢�⃗ 𝑡 𝒘 Visibility logical matrix 𝒘���⃗ 𝒑 Director vector of line 𝑝 𝒘𝒃 Beginning of the nominal time interval 𝒘𝒇 End of the nominal time interval 𝒘𝒔 Selected nominal interval 𝑿/𝒀 X relative to Y 𝒙,𝒚, 𝒛 Components of vectors

Greek Symbols 𝜶, 𝜷 Angles from the farthest points of the Sun's surface relative to the parallel to

the Earth 𝜶𝒕 Angle that determines the position of the satellite relative to the terminator

line 𝜸 Required half cone angle 𝜹 Angle of incidence of the sunlight from the center of the Sun relative to the

parallel to the Earth 𝜹𝑺𝑨𝑨𝑮𝒆𝒐 SAA region in the Geographical reference frame observed from ECI 𝜹𝟎𝑬𝑪𝑰 Initial value longitude of the SAA region in the ECI reference frame 𝜹𝟎𝑮𝒆𝒐 Initial value of the longitude of the SAA region in the Geographical

reference frame Δ𝝕 Increase in the longitude of the ascending node ΔΩ Increase in the longitude of the ascending node 𝝀 Parameter of line 𝑠 𝝁 Parameter of line 𝑝 𝝕 Argument of perigee �̇� Argument of perigee variation 𝜽 True anomaly 𝜽 Actual half cone angle 𝜽𝒕 Parameter of the terminator circle 𝝉𝒅 Transit duration 𝝉𝒆𝒇𝒇 Effective time interval duration 𝝉𝒏 Nominal duration of the observations �̇� Longitude of ascending node variation 𝝎𝑬 Rotation speed of the Earth 𝜴𝟎 Longitude of ascending node

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Chapter 1: Introduction

1.1 Thesis Objective

The main objective of this Thesis is to optimize the observations schedule of exoplanets

from a satellite, taking into account the different constraints that affect the observations visibility,

which are mainly the Earth, the Moon and the Sun.

1.2 Observing the sky from orbit

Currently, we have the possibility to observe the universe not only from the Earth, but

also from space. Observation satellites such as space telescopes allow us to reach much more

precision in the observations because they are free from the Earth's atmosphere, which distorts

and blocks light from the cosmos [1].

On the other hand, having an observation satellite in orbit requires a good planning and

pointing system because we have to consider the lack of visibility in certain instants produced

by eclipses, and the damage caused to the sensor because of the Sun's radiation, amongst

others.

1.2.1 Exoplanets

An exoplanet is a planet that orbits a star other than the Sun [2]. These planets are an

important field of study because, although we already know about the existence of many

exoplanets, we are still not able to determine accurately their structure and establish a precise

classification of all their properties [3].

Planetary transit surveys are one way of exoplanets detection that consists in the record

of the stars that are partially hidden during a certain amount of time due to the movement of an

exoplanet around them. Transit surveys made from ground have shown that the first detected

giant exoplanets are very different from those in our Solar System, they are located very close

to their host stars (more or less 0.1 AU). Also, some have non-coplanar orbits with the

equatorial plane of their host stars, and retrograde orbits are more common that it was

expected. This lead to the development of some theories, like planetary migration (see [4] for

more information) to explain this phenomena, which could be confirmed by precise

observations [3].

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The observation methods also depend on the size of the exoplanet that is going to be

observed. Observations from ground are limited to giant exoplanets for two main reasons.

First, the motions of the atmosphere are constantly bending the rays of light from each

star, which makes them appear to twinkle. This change of brightness can be reduced to one

part in 1000, which could be acceptable to detect giant exoplanets but not good enough to find

Earth-size planets.

Second, the detection of short transits requires a continuous observation, which means

to have a lot of telescopes around the globe dedicated to observe short transits considering the

position of the Moon, the Sun, and the weather conditions. This contingency would mean a very

expensive operation.

These reasons make necessary to employ space telescopes in order to observe and

characterize Earth-type planets more precisely1

.

1.2.2 Planetary transits and observation methods

The observation of exoplanets transiting bright stars (Visual Magnitude less than 12) will

allow us to get more information about their key parameters, like mass and radius, in order to

make a good characterization. For example, an accurate spectroscopic analysis will allow us to

know the size of the exoplanet, and its atmosphere composition and dynamics [3].

The number of transit surveys that take place from both ground and space, makes

necessary to classify exoplanets and highlight those more relevant for research [3]. Mainly,

transit surveys can be classified in two categories [5]:

1) Deep surveys with small pixel size: these surveys are focused on fainter stars but

cannot cover a wide part of sky.

2) Wide surveys with large pixel size: they can cover a wide area of the sky but cannot see

fainter stars. They are focused on brighter stars.

1.2.3 Relevant missions

There are several missions related to stars and exoplanets observation. In the following

subsections, we are going to review some of the past, present and future missions.

Some of the most important space telescopes for sky observation are the Spitzer, the

Hubble Space Telescope, and the James Webb Space Telescope that will be launched in 2018. 1 For more information, see http://kepler.nasa.gov/Mission/faq/#b1

3

It is expected that the James Webb Space Telescope will provide many observations of all kind

of astrophysical events, as the other two mentioned telescopes have already done.

However, discoveries made in the field of exoplanets by these kinds of telescopes have

been limited to large exoplanets in odd orbits. This is due to the high observing time needed to

follow the transits with the required precision. Also, their small field of view and the available

techniques and instruments make transit observations very expensive [5].

There are also smaller missions which provide less precision but full sky visibility, unlike

the telescopes named before. For example, the Chinese Hard X-ray Modulation Telescope

(HXMT) which performs a hard X-ray survey with all-sky visibility2

Amongst the existing exoplanets observation missions, there are already two satellites

focused on exoplanet transits, CoRoT and Kepler. These satellites have been gathering a lot of

important information about exoplanets during the last years. CoRoT and Kepler were designed

to stare at a certain area of the sky in order not to miss a transit and to observe the larger

number of them, focused on faint stars with visual magnitude between 13 and 16. In order to

characterize accurately an Earth size exoplanet, it is needed to observe brighter stars with a

visual magnitude typically between 6 and 9 [3].

.

Comparing Kepler and CoRot's mission characteristics, we can point out the importance

of staring at a target for a long time in order to obtain accurate photometry needed for the

detection of transits in faint stars3

After the success of Kepler observing transits in faint stars, next missions from ESA and

NASA are going to focus on Earth-like exoplanets transiting bright stars.

. Another important feature is that Kepler does not point

anywhere in the sky, it is pointing to only one fixed large area of the sky in the constellations

Cygnus and Lyra. This area was selected mainly because it can be seen continuously during

the mission and it is rich in stars similar to those Kepler can observe.

The most important future missions, which will have complementary goals, are:

- CHEOPS, ECHO and PLATO, from ESA,

- TESS and FINESS, from NASA.

This work will be focused on the CHEOPS mission. CHEOPS satellite will be able to point to anywhere in the sky, providing key targets to the M-class missions ECHO (and FINESS,

its NASA's complementary), and PLATO (with TESS, its NASA's complementary). PLATO, as

Kepler, will not be able to point at a given location on the sky.

2 For more information, see http://www.hxmt.cn/english/ 3 For more information, see http://cheops.unibe.ch/index.php/science/corot-and-kepler-vs-cheops

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1.3 Sky visibility and planning problem

1.3.1 Introduction to the problem

There are many constraints that affect the sky visibility of a star from a satellite, and

hence the exoplanets visibility. These constraints can be caused by two main reasons, physical

obstructions or loss of visibility:

• The satellite structural elements can produce obstructions in the line of sight of the

satellite.

• The Sun and the Moon's light reduces the visibility of the targets.

• The Earth also causes an obstruction and the light reflected on it decreases the

visibility.

• In low orbits (LEO), the South Atlantic Anomaly produce an amount of radiation that can

damage the sensor if it is on, and the ram atomic oxygen causes erosion in the primary

structures.

All these constraints are also strongly dependant on the satellite's orbit, architecture, and

attitude [6].

We will have to consider the main constraints that affect the satellite studied in order to

maximize some parameters, for example the observation time of the selected targets. This is

why observation planning is also an important problem to be considered. The bigger the area

and the number of targets we have to observe, the more important it will be to have a precise

target scheduling. Sky visibility and scheduling problems especially concern CHEOPS mission

because it is designed to point anywhere in the space to many specific targets [3].

In the next subsection we will review solutions from other missions in order to facilitate

the comprehension of the problem.

1.3.2 Previous solutions

All the space telescopes - like those named in section 1.2.3 Relevant missions - have

to deal with the sky visibility problem. Analyzing how this problem was solved for other related

missions will help us to understand it better and acquire know-how.

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From [6] and [7], we can deduce the importance of having a good representation and

planning of the observations to visualize the problem. The common way of representation is to

project the celestial sphere on a plane and then represent the constraints and targets. The sky

visibility can be represented considering the total observation time or time intervals, depending

on what we want to optimize. We can also represent the number of targets as a function of time

in order to know how many of them are visible considering the constraints.

Each constraint affects visibility in a different way and can be modeled in 3D as a cone

with the apex in the satellite involved instrument. For example, the most important constraint of

the HXMT satellite is the occultation by the Earth, which decreases considerably the total

observation time because of its 67º half cone angle and its period. In this case, the Moon is not

negligible because most of the targets are affected by it [7].

From [8], we can see how the scheduling optimization problem was solved for the

ROSAT mission. The mathematical method used consists of an algorithm and a heuristic

solution to a Generalized Assignment Problem (GAP). The mathematical model has the

following data as an input:

- Series of objects, lengths and priorities.

- Time intervals or slots.

- Visibility intervals, which are the intervals when the object can be observed, and it can

be obtained combining the above inputs with the visibility constraints.

- Adjustment telescope time to go from one object to another.

In this example, the output of the algorithm is the observations length required for the schedule.

The priority of the objects and the sensor adjustment time can be implemented by a

cost function depending on the most important parameters considered, for example the

observation time.

1.3.3 CHEOPS mission summary

In this next subsection, we will present the main objectives, relevant requirements and

constraints of the CHEOPS mission.

The main objectives of the mission are [9]:

1) To detect Earth-size planets transiting G5-type stars with a radius of 0.9 that of the Sun,

between the 6th and the 9th magnitudes in the V band, and with a signal-to-noise ratio

of 10. The depth of this transit is 100 parts-per-million. This depth requires a

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photometric precision of 20 ppm in 6 hours of integration time which corresponds to the

transit duration of a planet with a revolution period of 50 days.

2) To provide precise radii measurements of Neptune-size planets with revolution periods

up to 13 days which are transiting stars between the 6th and the 13th magnitudes in the

V band. The samples with signal-to-noise ratios above 30 and radii between 1.5 and 6

Earth radius will have a precision 10% better than the achieved with the new generation

of ground-based transit searches.

3) To provide key targets for future ground and space-based facilities.

4) To study the energy transport in the atmosphere of Hot Jupiter type of planets.

The relevant specifications for the pointing and planning problem are [9]:

Concerning the orbit, it will be circular and Sun-synchronous with the following

characteristics:

- Altitude: approximately 800 km.

- Local time of ascending node (LTAN): 6 am or 6 pm (less favourable).

- Inclination: 98º.

- Orbital period: approximately 100 min.

- Launch date: end of 2017.

The main pointing requirements and constraints are:

- Observe one star at a time.

- Line of sight (LoS): CHEOPS should be able to point to any target contained in a 60º

half cone angle from the anti-Sun direction.

- During all mission phases after the opening of the telescope cover, the Sun must not

be in the Field of View of the telescope.

- Once opened, the telescope cover must not be closed.

- Unobstructed Field of View of 180º full cone aperture of the instrument.

- Pointing Stability: Half cone angle between actual and desired LoS directions, Absolute

Performance Error (APE):

o < 8 arcsec at 68 % confidence using temporal statistical interpretation over a

10 hour observing period.

o < 1 arcmin at 95% confidence, using the temporal statistical interpretation prior

to the start of observations, to be confirmed (TBC).

- The spacecraft shall maintain the pointing during interruptions of the observations.

- The Attitude and Orbital Control Systems (AOCS) is 3 axis established and Nadir

locked.

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The sky visibility requirements and constraints are the key parameters to define the

visibility and planning problem.

The sky visibility requirements are:

- 25% of the sky (with 2/3 in the southern hemisphere) for a minimum cumulative duration

of 13 days per year and per target (goal 15 days per year), and a minimum accessible

time per orbit and per target of 80% of the orbital period (>80 min for 100-min

spacecraft orbital period).

- 50% of the sky for a minimum consecutive duration of 50 days per year and per target

(with the goal of 60 days) and a minimum accessible time per orbit and per target of

50% of the orbital period (>50 min for 100-min spacecraft orbital period).

The sky visibility constraints are:

- Observation interval: the nominal observation time interval for the NGTS-coded targets

is 12 hours and 48 hours (28 CHEOPS' orbits) for the rest of the targets. The definition

of these nominal intervals is assumed to be continuous and centered on the transit.

However, the target observations carried out during these intervals can be

discontinuous.

- The Sun's exclusion angle: the Sun must not be inside a cone around the line-of-sight

of the telescope having a half-angle of at most 120 degrees.

- The occultation by the Earth: the target shall not be occulted by the Earth, taking a

slightly enhanced Earth radius by 100 km to avoid atmospheric glow.

- The Earth's stray light exclusion angle: the illuminated surface of the Earth (facing the

satellite) shall not be inside a cone around the line-of-sight of the telescope having a

half-angle of 35 degrees (goal: 28 deg).

- The Moon's exclusion angle: the bright Moon shall not be inside a cone around the line-

of-sight of the telescope having a half-angle of 5 degrees.

- The South Atlantic Anomaly: the spacecraft shall be outside the South Atlantic Anomaly

during science observations (< 2 particles/s/cm2).

Other specifications to be considered:

- Mission nominal lifetime: 3.5 years (can be extended to 5 years).

- Transit-to-noise ratios:

o S/Ntransit = 5 to obtain a clear detection.

o S/Ntransit = 30 to obtain the radius of the planet with precision.

8

1.4 Goals of the thesis

In this thesis, we will explain the procedures to reach two main goals:

• First, we will calculate the visibility windows for the CHEOPS satellite. In order to do

this, we will develop a computational program to identify CHEOPS' orbit and to combine

it with the most important visibility constraints, see Chapter 2: Visibility computation.

• Then, we will define the mission planning, optimizing some parameters such as the

effective time by means of cost functions and artificial intelligence algorithms, see

Chapter 3: Mission Planning.

9

Chapter 2: Visibility computation

In this part of the thesis, we will calculate the targets visibility windows considering the

diverse constraints. Also, we will define the visibility windows as the periods of time when the

target is visible. As we will see in section Chapter 3: Mission Planning, the visibility windows will

be very useful for the optimization process.

We will make use of MATLAB R2011a to develop most of the calculations in this thesis.

The reference frame we are going to use is the J2000 Earth-centered inertial, but centered in

the satellite instead of the center of the Earth. We can set the time reference from the estimated

launch date (see subsection 1.3.3 CHEOPS mission summary). Finally, as CHEOPS is

expected to be launched at the end of 2017, we will set the time reference for the beginning of

the observations on January 1st, 2018 at 11:00 AM UTC.

2.1 Definition of targets

The targets of the CHEOPS mission consist of 201 stars which are likely to host

planets, because it is known that they have a transit with a determined period at a certain

epoch.

As the precision required is very high and the satellite can only point to one star at a

time, we will model the pointing as a vector directed straight to the star. This error is in the side

of safety, because we are assuming that the aperture of the sensor is the minimum possible

and the APE must be zero (see section 1.3.3 CHEOPS mission summary).

Also, we can neglect the difference between centering the reference frame in the center

of the Earth or in the satellite, because the targets are at a very large distance. So, we can

create the pointing vector directly from the list of targets given the latitude and longitude.

2.2 Orbit propagation

Next, we will define the orbit and its propagation in order to identify where the satellite is

in each moment.

As we can observe in section 1.3.3 CHEOPS mission summary, CHEOPS' orbit must

be Sun-synchronous with a preferable local time of ascending node of 6:00 AM. A Sun-

synchronous orbit is a particular kind of Low Earth Orbit (LEO) that combines altitude and

10

inclination so that the satellite ascends or descends over an Earth latitude at the same local

mean solar time.

Sun-synchronous orbits are very important for scientific observations because the angle

between the Sun and the Earth’s surface below the satellite remains relatively constant. This

characteristic allows scientists to compare images from the same season over several years

without worrying too much about changes in shadows and lighting areas.

Another characteristic is that Sun-synchronous orbits are nearly polar and retrograde,

which means that the satellite is orbiting the Earth in the opposite direction to the Earth's

rotation, see Figure 2.1.

Also, as the local time of ascending node is 6:00 AM (sunrise) the orbit will be dawn

Sun-synchronous, a special case of Sun-synchronous orbit where the satellite passes over the

equator ridding the line which separates day and night, called terminator line4

.

Figure 2.1: Sun-synchronous orbit5

As CHEOPS is not launched yet, the orbital parameters are not completely fixed.

Therefore, we will base our calculations on the orbital characteristics explained before and the

classical orbital elements of SMOS, a satellite with similar orbit. We will do some corrections

and adjust the argument of perigee and the longitude of ascending node so that the perigee

passage time, which will correspond to the reference time, i.e. January 1st, 2018 at 11:00 AM

UTC (Coordinated Universal Time), matches requirement of a 6:00 AM Sun-synchronous orbit,

see Table 2.1.

4 For more information, see http://earthobservatory.nasa.gov/Features/OrbitsCatalog/page2.php 5 Illustration by R. Simmon obtained from http://earthobservatory.nasa.gov/Features/OrbitsCatalog/page2.php

11

Classical Orbital Parameters

Semi-major axis (𝒂) 7125 km

Eccentricity (𝒆) 0.00008 (~ circular orbit)

Perigee passage time (T0) 2018 JAN 01 11:00:00UTC

Inclination (𝒊) 98º

Longitude of ascending node (𝜴𝟎) 192º

Argument of perigee (ϖ) 48º

Table 2.1: Classical orbital parameters for CHEOPS' orbit

We can derive other useful parameters from the classical orbital elements:

- Mean orbital velocity (𝑛): 𝑛 = �𝜇⊕𝑎3

= 0.001 𝑠−1.

- Orbital period (𝑇): 𝑇 = 2𝜋�𝜇⊕

.𝑎32 = 99.743 𝑚𝑖𝑛.

- Longitude of ascending node variation �Ω̇�: Ω̇ = −3𝑛𝐽2𝑅⊕

2

2𝑎2(1−𝑒2)2cos 𝑖 = 0.9771 °/𝑑𝑎𝑦.

- Increase in the longitude of the ascending node (ΔΩ): ΔΩ=Ω0 + Ω̇(𝑡 − 𝑡0).

Where 𝐽2 is a constant that depends on the shape of the celestial body, 𝑅⊕ is the medium

radius of the Earth6

Once we have calculated the orbital parameters, we have to set the time sampling. As

the observations have to be done during three and a half years, we will set the time intervals

depending on two parameters:

, 𝜇⊕ is the standard gravitational parameter of the Earth, 𝑡 is a general

instant of time in Ephemeris Time (ET) seconds, and 𝑡0 is the reference time in ET seconds.

- The time interval division in during one orbit (𝑇𝑑𝑖𝑣)

- The number of orbits (𝑞𝑑𝑖𝑣).

Next, we will calculate the propagation of the orbit by means of Kepler's equation for

circular orbits,

𝜃 ≈ 𝑀 (2.1).

6 From now on, we will suppose that the Earth is a sphere of radius 𝑅⊕ = 6378km.

12

Where 𝑀 is the mean anomaly and 𝜃 the true anomaly. We can assume this hypothesis

because by comparing the mean anomaly and the eccentric anomaly we obtain that the

deviation is 𝑂(10−6).

The mean anomaly can be calculated with the formula (2.2):

𝑀 = 𝑛∆𝑡 (2.2).

From (2.1) and (2.2) we can easily obtain the true anomaly for an instant of time.

Given the classical orbital parameters and the time, we can obtain the position vector of

the satellite for each instant of time in the Earth-centered J2000 reference frame. The way to

proceed is:

• First, to calculate the position vector in the orbit reference frame centered in the

focus (F).

• Then, to execute three axis rotations to obtain the position vector in the J2000 reference

vector (R).

To perform these calculations we will use the next equations,

𝑟𝑠𝑎𝑡/𝐸𝐹 =𝑝

1 + 𝑒 𝑐𝑜𝑠𝜃�𝑐𝑜𝑠 𝜃𝑠𝑖𝑛 𝜃

0� (2.3).

𝐶𝑅𝐹(𝜔, 𝑖,Ω) = �𝑐Ω𝑐𝜛 − 𝑠Ω𝑠𝜛𝑐𝑖 𝑠Ω𝑐𝜛 + 𝑐Ω𝑠𝜛𝑐𝑖 𝑠𝜛𝑠𝑖−𝑐Ω𝑠𝜛 − 𝑠Ω𝑐𝜛𝑐𝑖 −𝑠Ω𝑠𝜛 + 𝑐Ω𝑠𝜛𝑐𝑖 𝑐𝜛𝑠𝑖

𝑠Ω𝑠𝑖 −𝑐Ω𝑠𝑖 𝑐𝑖�

(2.4).

𝑟𝑠𝑎𝑡/𝐸𝑅 = �𝜇𝑝

(𝐶𝑅𝐹)𝑇 �cos𝜃𝑠𝑒𝑛 𝜃

0�

(2.5).

Where 𝑠𝜙 means sin𝜙, 𝑐𝜙 means cos𝜙, 𝑟𝑠𝑎𝑡/𝐸𝐹 is the position vector of the satellite in the orbit

reference frame (F), 𝑝 is the semi-latus rectum, 𝜃 is the true anomaly, 𝐶𝑅𝐹 is the rotation matrix,

and 𝑟𝑠𝑎𝑡/𝐸𝑅 is the position vector of the satellite in the J2000 reference frame (for a detailed

theoretical explanation see [10]).

We will implement these ideas in a MATLAB script, using also the MATLAB version of

the NASA's JPL software, SPICE [11]. This software contains a large amount of astronomical

data such as the ephemeris of planets and stars.

13

Plotting the result for an instant of time, we obtain the satellite orbit in three dimensions

as in Figure 2.2. Then, converting from Cartesian to latitude-longitude coordinates, we obtain

the satellite orbit in the celestial sphere, see Figure 2.3.

Figure 2.2: CHEOPS' orbit 3D in 𝒕𝟎

a = 7125 km

e = 0.00008

i = 98º

Ω = 192º

ϖ = 48º

T0 = January 1st, 2018

CHEOPS' Orbit 3D

14

Figure 2.3: CHEOPS' orbit 2D in 𝒕𝟎

Finally, we will confirm the results observing Figure 2.2 and Figure 2.3, which represent

the satellite position for the reference time 𝑡0. To confirm that the orbit is 6:00 AM local mean

solar time, the satellite should be passing over the terminator at 𝑡0 because we defined 𝜛 for

𝑡0 = 𝑇0.

In Figure 2.2, the orange line represents the vector that points to the Sun, the red line

represents CHEOPS's orbit, and the yellow and blue areas are the Earth's day and night

regions, respectively. As the reference frame we are using is the Earth Centered Inertial (ECI),

the Earth rotates clockwise, so at 𝑡0 CHEOPS is moving over the terminator from the dark to the

illuminated side of the Earth7

We can obtain similar deductions from

. Also, the satellite is on the 𝑥𝑦 plane, which means that it is over

the Earth's equator at sunrise. This situation should be repeated every 𝑡0.𝑛𝑇 periods of time in

order to determine a 6:00 AM sunrise Sun-synchronous orbit.

Figure 2.3, we can observe that the satellite is

exactly on the equator at 𝑡0. In addition, we can check that the orbit is almost polar because it is

close but does not reach 90º and -90º.

7 We will explain the terminator and the illuminated and dark areas calculation in subsection 2.3.5 The Earth's stray light constraint

-180 -135 -90 -45 0 45 90 135 180

-90

-45

0

45

90CHEOPS` Orbit

Longitude (degrees)

Latit

ude

(deg

rees

)

-90

-45

0

45

90

-180 -135 -90 -45 0 45 90 135 180

15

2.3 Constraints and visibility calculation

In this subsection, we will define the visibility of the targets, then we will explain and

calculate the constraints that affect the visibility of the targets, and finally we will characterize

the visibility windows taking into account all the constraints.

To calculate the constraints, we will divide the visibility problem considering each

constraint independently. This division of the problem is useful for many reasons.

• First, it allows us to simplify the problem and to avoid concatenated errors.

• Second, it lets us identify which constraint is causing visibility problems.

• Last, it means a time calculation improvement because of the two previous reasons. For

example, in case of large and precise calculations, we can use several computers to

obtain the results. Calculation time is very important for this thesis since we have to do

calculations for many targets, with many constraints, and with small time intervals, see

subsection 3.1.2 Class of complexity and the importance of the visibility windows.

In order to do the visibility characterization, first we will calculate and represent the

effect of the constraint in the spacecraft centered celestial sphere, and then we will calculate the

visible periods for each target and instant of time considered. Finally, we will illustrate this

section with some examples of visibility windows defined for several targets and an arbitrary

mission time.

2.3.1 Targets transit visibility

The transits of the targets happen only at a specified interval of time, which is given in a

target list. CHEOPS will have to observe each target amongst certain time intervals in order not

to miss any transit, considering useless any observation produced when the transits are not

happening. Therefore, we will have to calculate first when the transits are happening, and then

establish the time interval required to do the observations.

The transits are defined by three parameters:

- The instant of time when the transit was first detected (𝑡𝑐), we will assume that

these detections were made in average during half of the duration of the transit.

- The period of the exoplanet revolution around the host star (𝑃𝑡).

- The transit duration (𝜏𝑑), which is not that important because the duration of the

observations will be provided by the nominal time intervals duration (𝜏𝑛), which we

will explain in the next lines.

16

As the mission will start some time after the transit was discovered, we need to adjust 𝑡𝑐, and

the beginning of the observation mission (𝑡0). Next, we will determine the time instants centered

on the transit, when the transit occurs for all the mission (𝑡𝑐𝑖,𝑛𝑝), for the target 𝑖 and number of

the exoplanet revolutions around the host star 𝑛𝑝 considered, see Figure 2.4. This setting can

be implemented with a simple algorithm:

1. Add 𝑛𝑝𝑡.𝑝𝑡 periods to 𝑡𝑐, being 𝑛𝑝𝑡 = (1, 2, … ,𝑛).

2. Increase the number of 𝑛𝑝𝑡 until 𝑡𝑐 + 𝑛𝑝𝑡.𝑃𝑡 ≥ 𝑡0 and save that value as 𝑛𝑝0 = 𝑛𝑝𝑡.

3. Define 𝑡𝑐𝑖,0, as 𝑡𝑐𝑖,0 = 𝑡𝑐 + 𝑛𝑝0.𝑃𝑡.

4. Add 𝑛𝑝.𝑝𝑡 periods to 𝑡𝑐𝑖,0 to determine each 𝑡𝑐𝑖,𝑛𝑝 during the mission time, being

𝑛𝑝 = (1, 2, … ,𝑚).

5. Finish when 𝑡𝑐𝑖,0 + 𝑛𝑝.𝑃𝑡 ≥ 𝑡𝑓

Where 𝑡𝑐𝑖,0 is the instant of time when the first transit occurs for the target 𝑖 considered, and 𝑡𝑓 is

the end of the mission instant of time, nominally 𝑡𝑓 = 𝑡0 + 3.5 𝑦𝑒𝑎𝑟𝑠.

Figure 2.4: Observation interval definition

Once we have calculated when the transits are happening for all the mission, we can

determine the nominal intervals of time when we can obtain useful observations by means of

two parameters (as we can see in Figure 2.4):

- The time instants when the transits occur (𝑡𝑐𝑖,𝑛𝑝) , previously calculated.

- The nominal duration of the observations (𝜏𝑛𝑖 ), determined by CHEOPS requirements,

see 1.3.3 CHEOPS mission summary.

The nominal intervals of observation establish the margins of time when the targets

should be observed, and it is defined as a continuous period of time of either 12 or 48 hours,

which depends and is centered on the target [3]. To determine the nominal intervals of

observation, we will calculate the beginning (𝑡𝑖,𝑛𝑝−1) and end (𝑡𝑖,𝑛𝑝+1) of the nominal

observation interval as 𝑡𝑖,𝑛𝑝−1 = 𝑡𝑐𝑖,𝑛𝑝 −𝜏𝑛𝑖2

and 𝑡𝑖,𝑛𝑝+1 = 𝑡𝑐𝑖,𝑛𝑝 +𝜏𝑛𝑖2

, see Figure 2.4.

time

𝑡𝑐𝑖,𝑛𝑝 𝑡𝑐𝑖,𝑛𝑝+1

𝑡𝑐𝑖,𝑛𝑝−1

τ𝑛𝑖 /2

τ𝑛𝑖 /2

17

Also, we have to consider that the observations do not have to be strictly continuous

during the nominal time interval, because of the constraints which will interfere with the targets

visibility. With this, we will define the effective time interval (𝜏𝑒𝑓𝑓) which will be the amount of

time that the stars will be visible during the nominal time. Ideally, we would like to have

𝜏𝑒𝑓𝑓 = 𝜏𝑛.

In conclusion, the nominal time intervals represent the instants of time for useful

observations in case we have continuous visibility of the target (which are represented in Figure

2.5), and the effective time intervals the real amount of the observable time considering the

constraints.

In the next subsections, we will explain and calculate the effect that the constraints

produce in the visibility of the targets, and hence the reduction of the effective observation

interval.

18

Figure 2.5: Nominal interval of observation for 50 targets and 1.5 months

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950

Cal

enda

r Tim

e (Y

YYY

MM

M D

D H

H:M

M:S

S)

Target Code Number

2018 MAY 01 11:00:002018 MAY 02 03:37:262018 MAY 02 20:14:522018 MAY 03 12:52:172018 MAY 04 05:29:432018 MAY 04 22:07:092018 MAY 05 14:44:352018 MAY 06 07:22:012018 MAY 06 23:59:262018 MAY 07 16:36:522018 MAY 08 09:14:182018 MAY 09 01:51:442018 MAY 09 18:29:102018 MAY 10 11:06:352018 MAY 11 03:44:012018 MAY 11 20:21:272018 MAY 12 12:58:532018 MAY 13 05:36:192018 MAY 13 22:13:442018 MAY 14 14:51:102018 MAY 15 07:28:362018 MAY 16 00:06:022018 MAY 16 16:43:282018 MAY 17 09:20:532018 MAY 18 01:58:192018 MAY 18 18:35:452018 MAY 19 11:13:112018 MAY 20 03:50:372018 MAY 20 20:28:022018 MAY 21 13:05:282018 MAY 22 05:42:542018 MAY 22 22:20:202018 MAY 23 14:57:462018 MAY 24 07:35:112018 MAY 25 00:12:372018 MAY 25 16:50:032018 MAY 26 09:27:292018 MAY 27 02:04:552018 MAY 27 18:42:202018 MAY 28 11:19:462018 MAY 29 03:57:122018 MAY 29 20:34:382018 MAY 30 13:12:042018 MAY 31 05:49:292018 MAY 31 22:26:552018 JUN 01 15:04:212018 JUN 02 07:41:472018 JUN 03 00:19:132018 JUN 03 16:56:382018 JUN 04 09:34:042018 JUN 05 02:11:302018 JUN 05 18:48:562018 JUN 06 11:26:222018 JUN 07 04:03:472018 JUN 07 20:41:132018 JUN 08 13:18:392018 JUN 09 05:56:052018 JUN 09 22:33:302018 JUN 10 15:10:562018 JUN 11 07:48:222018 JUN 12 00:25:482018 JUN 12 17:03:142018 JUN 13 09:40:392018 JUN 14 02:18:052018 JUN 14 18:55:312018 JUN 15 11:32:57

Nom

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inte

rval

of o

bser

vatio

n fo

r 50

targ

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and

1.5

mon

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19

2.3.2 Constraint of the occultation by the Earth

As the satellite is orbiting the Earth, there are some instants during the course when the

Earth is between the satellite and the target. This eventuality occults the target from the satellite

sensor making it not visible.

Since the orbit is very low, we can predict that this constraint is going to be critical

because the Earth is going to hide a large part of the sky.

To model this constraint, we will have to make some geometrical calculus. First, we will

calculate the Earth's eclipse half cone angle (𝛾𝐸𝑜𝑐𝑐), which defines the occultation by the Earth

region, see Figure 2.6.

Figure 2.6: Eclipse half cone angle (𝜸𝑬𝒐𝒄𝒄) definition, not to scale

As we can observe in Figure 2.6, we can calculate 𝛾𝐸𝑜𝑐𝑐 by solving the right triangle

formed by the 𝑅⊕ plus 100 km to avoid the atmospheric glow, its tangent from the satellite, and

the radius of the orbit, 𝑎. The result is 𝛾𝐸𝑜𝑐𝑐 = 64.5 °.

Since we know the position vector of the satellite respect to the Earth in the ECI

reference frame �𝑟𝑠𝑎𝑡/𝐸�, we can obtain the position vector of the Earth in the satellite centered

ECI reference frame �𝑟𝐸/𝑠𝑎𝑡� simply by changing its sign.

The vector 𝑟𝑠𝑎𝑡/𝐸 and the angle 𝛾𝐸𝑜𝑐𝑐 define a cone in the space which determines the

region of the sky where the targets are hidden from the LoS of the satellite because the Earth is

in between. This occultation region can be represented in the spacecraft centered celestial

sphere, see Figure 2.7.

𝛾𝐸𝑜𝑐𝑐

𝑎

𝑅⊕ CHEOPS

20

Figure 2.7: Spacecraft centered celestial sphere considering the constraint of the occultation by the Earth. Case 1. The Earth's occultation region is

represented by a blue circle

In Figure 2.7, we can see the area affected by the occultation due to the Earth

represented by a blue circle and all the targets represented by green asterisks.

As we want to determine which are the targets that are affected by the constraint, we

can use the MATLAB function inpolygon, which will gives us as an output the points inside a

certain polygon. In this case, we define the polygon as the circle that represents the occultation

region. The output of this function are the hidden targets in the instant of time considered, which

are represented by red squares in Figure 2.8.

-180 -135 -90 -45 0 45 90 135 180

-90

-45

0

45

90

RA (degrees)

dec

(deg

rees

)

Spacecraft centered celestial sphere (Constraint of the occultation by the Earth)

-90

-45

0

45

90

-180 -135 -90 -45 0 45 90 135 180

21

Figure 2.8: Spacecraft centered celestial sphere considering the constraint of the occultation by the Earth. Case 2. The Earth's occultation region is represented by a blue circle, the visible targets by green stars and the

hidden targets by red squares

It is important to consider in the algorithm that the celestial sphere is limited vertically by

[-90º,90º] and horizontally by [-180º,180º], so we will add and subtract 360º if the right

ascension (RA) of the occultation region is lower than -180º or higher than 180º. Similarly, we

will add and subtract 180º when the declination is lower than -90º or higher than 90º. Otherwise,

we would not be considering targets that should be hidden. That is why in Figure 2.8 one part of

the occultation region is in the upper area of the graph and the other part in the lower area.

Figure 2.7 and Figure 2.8 correspond to two different instants of time. If we plot the

results for many consecutive instants of time, we can make a movie considering each plot one

frame of the movie, where we can observe that the Earth is moving from RA=180º to RA=-180

because the orbit is retrograde.

With these calculations, we can obtain the visibility of each target for each instant of

time, which will allow us to define the visibility windows8

The exclusion region calculation method explained allows us to understand the

constraint better and to perceive the visibility problem in an intuitive way, letting us observe the

sky from the satellite point of view. However, this method is computationally slow since we have

to calculate all the perimeter of the occultation region. So, we will develop a more efficient

.

8 Notice that we are not considering the nominal time intervals in these calculations yet. Therefore, this is the visibility of the target stars without considering if a transit is happening or not.

-180 -135 -90 -45 0 45 90 135 180

-90

-45

0

45

90

RA (degrees)

dec

(deg

rees

)Spacecraft centered celestial sphere (Constraint of the occultation by the Earth)

-90

-45

0

45

90

-180 -135 -90 -45 0 45 90 135 180

22

model in terms of calculation time (see Table 3.1 in section 3.1.2 Class of complexity and the

importance of the visibility windows to check the calculation time comparison).

The alternative method consists in the calculation of the visibility half cone angle (𝜃𝐸𝑜𝑐𝑐)

and the Earth's eclipse half cone angle (𝛾𝐸𝑜𝑐𝑐), see Figure 2.9, and then to compare both angles

stating the next visibility criterion:

• If 𝜃𝐸𝑜𝑐𝑐 > 𝛾𝐸𝑜𝑐𝑐 the target is visible in the time instant considered.

• If 𝜃𝐸𝑜𝑐𝑐 < 𝛾𝐸𝑜𝑐𝑐 the target is occulted by the Earth in the time instant considered.

Figure 2.9: Visibility criteria for the constraint of the occultation by the Earth, comparison between 𝜽𝑬𝒐𝒄𝒄 and 𝜸𝑬𝒐𝒄𝒄, not to scale

As we can see in Figure 2.9, the Target 2 has an angle 𝜃𝐸𝑜𝑐𝑐2 lower than 𝛾𝐸𝑜𝑐𝑐.

Therefore, the target is not visible because the LoS intersects the Earth's occultation region.

However, Target 1 can be observed because its LoS does not intersect the Earth's occultation

region for the instant of time considered.

To develop the algorithm we will consider the next calculations and deductions. The

angle 𝜃𝐸𝑜𝑐𝑐 is defined by the LoS vector (𝑟𝐿𝑜𝑆) and the vector that points to the centre of the

Earth in the satellite reference frame (𝑟𝐸/𝑠𝑎𝑡). The vector 𝑟𝐸/𝑠𝑎𝑡 was previously calculated, and

the vector 𝑟𝐿𝑜𝑆 can be obtained directly from the coordinates of the targets, see section 2.1

Definition of targets. Finally, the angle 𝜃𝐸𝑜𝑐𝑐 can be calculated substituting 𝑟𝐿𝑜𝑆 and 𝑟𝐸/𝑠𝑎𝑡 in the

dot product definition:

cos𝜃𝐸𝑜𝑐𝑐 =𝑟𝐿𝑜𝑆 . 𝑟𝐸/𝑠𝑎𝑡

|𝑟𝐿𝑜𝑆|. �𝑟𝐸/𝑠𝑎𝑡� (2.6).

𝛾𝐸𝑜𝑐𝑐 𝜃𝐸𝑜𝑐𝑐1

𝜃𝐸𝑜𝑐𝑐2

Target 1

Target 2

𝑅⊕

CHEOPS

23

Then, we will define an if loop with the visibility criterion explained above, which creates

a logical matrix showing a value of 1 for the visible targets, and 0 the hidden targets. Using

loops to calculate the logical matrix for many targets and instants of time, we can define the

instants of time when the targets are hidden, as we can see in Figure 2.10 and Figure 2.11.

Figure 2.10: Time intervals when the satellite is occulted by the Earth for 50 targets and 1 orbit

Figure 2.11: Time intervals when the satellite is occulted by the Earth for 50 targets and 1.5 days

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Occultation by the Earth time intervals for 50 targerts and 1 orbit

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Occultation by the Earth time intervals for 50 targerts and 1.5 days

24

The red bars in Figure 2.10 and Figure 2.11 represent the time intervals when the target

star is occulted by the Earth. We can check these results using some Fermi estimations, which

are estimation problems based on approximations, dimensional analysis and identifying

assumptions. Fermi problems involve making justified guesses about quantities and their lower

and upper bounds. Also, we will compare the results using the videos based on the first

calculation method described in this section.

In the videos we can observe that the Earth hides most of the targets only once per

orbit, and the occultation period lasts a bit less than half of the orbit. Comparing with Figure

2.10, we can notice that there is only one continuous red bar per orbit, whose length is a bit

smaller than half of the orbit.

We can do a similar test with Figure 2.11. As the orbit period is more or less 100

minutes and the time considered is 1.5 days, dividing these two quantities we obtain that there

should be more or less 21 occultation intervals per target, which is the result we obtain if we

count the bars in Figure 2.11.

The last step to obtain the visibility window considering only the constraint of the

occultation by the Earth is to take into account the periods of time when the transits occur.

As we explained in section 2.3.1 Targets transit visibility, we obtain useful observations

when the transits are happening in each target, i.e. during the nominal time intervals. This

means that not all the white spaces in Figure 2.11 provide useful observations. In order to know

when the nominal time intervals are affected by the constraint of the occultation by the Earth,

we will combine the logical matrix that is represented in Figure 2.11 with the nominal time

intervals, as the represented in Figure 2.5. The result is shown in Figure 2.12.

25

Figure 2.12: Visibility window for the constraint of the occultation by the Earth for 50 targets and 1.5 days

Where the red bars represent the time intervals occulted by the Earth and the green bars

represent the effective time intervals, i.e. the observable time intervals within the nominal time

intervals.

To conclude this subsection, we will determine the percentage of sky visibility per

instant of time by calculating the area that the Earth's exclusion region covers in the celestial

sphere,

𝐻𝑖𝑑𝑑𝑒𝑛 𝑎𝑟𝑒𝑎 =𝜋. 𝛾𝐸𝑜𝑐𝑐2

𝑐𝑒𝑙𝑒𝑠𝑡𝑖𝑎𝑙 𝑠𝑝ℎ𝑒𝑟𝑒 𝑎𝑟𝑒𝑎. 100 = 20.2 %

(2.7).

𝑉𝑖𝑠𝑠𝑖𝑏𝑙𝑒 𝑎𝑟𝑒𝑎 = 100 − ℎ𝑖𝑑𝑑𝑒𝑛 𝑎𝑟𝑒𝑎 = 79.2 %

(2.8).

Where the celestial sphere area is 360º times 180º.

2.3.3 Constraint of the Moon's exclusion angle

As the Earth, the Moon is moving respect to the satellite centered reference frame, so it

could hide the target star from the LoS of the satellite. In addition, the Sun's light is reflected on

the Moon's surface, which can produce damage in the sensor. The procedures to model this

constraint are similar to the ones explained in subsection 2.3.2 Constraint of the occultation by

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Visibility window for the constraint of the occultation by the Earth (50 targets and 1.5 days)

Occulted by the Earth time intervalEffective time interval

26

the Earth, but in this case the Moon's half cone angle of the exclusion region is given in the

requirements, being 𝛾𝑀𝑜𝑐𝑐 = 5° (see subsection 1.3.3 CHEOPS mission summary).

To calculate the position of the Moon in the satellite centered reference frame we need

the Moon's ephemeris, which can be obtained using the SPICE software [11]. SPICE's function

cspice_spkpos gives the Moon's ephemeris in the Earth's centered J2000 reference frame

�𝑟𝑀/𝐸�. With this data and the position of the satellite 𝑟𝑠𝑎𝑡/𝐸, we can obtain the position of the

Moon respect to the satellite 𝑟𝑀/𝑠𝑎𝑡, see Figure 2.13.

Figure 2.13: Visibility criteria for the constraint of the occultation by the Moon, comparison between 𝜽𝑴𝒐𝒄𝒄 and 𝜸𝑴𝒐𝒄𝒄, not to scale

Similarly to subsection 2.3.2 Constraint of the occultation by the Earth, we can define an

exclusion cone with the half cone angle 𝛾𝑀𝑜𝑐𝑐 and the direction of the vector 𝑟𝑀/𝑠𝑎𝑡 . This half

cone angle defines the occultation region produced by the Earth, which can be represented in

the spacecraft centered celestial sphere, as we can see in Figure 2.14.

𝑟𝑀/𝐸

𝑟𝑀/𝑠𝑎𝑡

𝑟𝑠𝑎𝑡/𝐸

Earth

Moon

CHEOPS

𝑅⊕

𝛾𝑀𝑜𝑐𝑐

𝜃𝑀𝑜𝑐𝑐 𝑟𝐿𝑜𝑆

27

Figure 2.14: Spacecraft centered celestial sphere considering the constraint of the Moon's exclusion angle

Observing Figure 2.14 we can notice that the Moon's exclusion region produced is very

small, calculating the percentage of visible and hidden regions of the sky we obtain that the

99.9% of the sky is visible for each instant of time considering only the Moon's exclusion angle

constraint. Therefore, the Moon's constraint will only be more problematic in the areas of the sky

where the target stars are very close to each other. Also, the exclusion region produced by the

Moon does not move as much as the one produced by the Earth, since the spacecraft is not

orbiting the Moon and the Moon's period is about one month. Consequently, this constraint will

not produce any occultation most of the time.

Next, we will apply to this constraint the second method explained in 2.3.2 Constraint of

the occultation by the Earth, which consists in the comparison between the angle 𝛾𝑀𝑜𝑐𝑐, and the

angle 𝜃𝑀𝑜𝑐𝑐 formed by the vectors 𝑟𝑀/𝑠𝑎𝑡 and 𝑟𝐿𝑜𝑆, (see Figure 2.13).

We will apply the same visibility criterion than in 2.3.2 Constraint of the occultation by

the Earth, obtaining a logical matrix with the hidden time intervals, which is represented in

Figure 2.15.

-180 -135 -90 -45 0 45 90 135 180

-90

-45

0

45

90

RA (degrees)

dec

(deg

rees

)Spacecraft centered celestial sphere (Constraint of the occultation by the Moon)

-90

-45

0

45

90

-180 -135 -90 -45 0 45 90 135 180

28

Figure 2.15: Time intervals when the satellite is affected by the Moon's exclusion region for all the targets and 6.5 days

Finally, we will combine the hidden time intervals with the nominal time intervals to

obtain the effective time intervals, see Figure 2.16. In Figure 2.15 and Figure 2.16 we can check

that the constraint of the Moon's exclusion angle only affects a few targets, and that this

constraint is considerably less important than the occultation by the Earth.

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:59:

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Occultation by the Moon time intervals for all the targets and 6.5 days

29

Figure 2.16: Visibility window for the constraint of the Moon's exclusion angle for all the targets and 6.5 days

05101520253035404550556065707580859095100

105

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MM

M D

D H

H:M

M:S

S)

Target Code Number

2018 JAN 01 11:00:00

2018 JAN 01 15:09:21

2018 JAN 01 19:18:43

2018 JAN 01 23:28:04

2018 JAN 02 03:37:26

2018 JAN 02 07:46:47

2018 JAN 02 11:56:09

2018 JAN 02 16:05:30

2018 JAN 02 20:14:52

2018 JAN 03 00:24:13

2018 JAN 03 04:33:34

2018 JAN 03 08:42:56

2018 JAN 03 12:52:17

2018 JAN 03 17:01:39

2018 JAN 03 21:11:00

2018 JAN 04 01:20:22

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2018 JAN 04 09:39:05

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2018 JAN 04 22:07:09

2018 JAN 05 02:16:30

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2018 JAN 06 07:22:01

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2018 JAN 06 15:40:43

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2018 JAN 06 23:59:26

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)

30

2.3.4 Constraint of the Sun's exclusion angle

The Sun's radiation can blind the satellite sensor making it unable to observe the

targets. This constraint will be the driver for the visibility calculation, because in CHEOPS'

requirements (see section 1.3.3 CHEOPS mission summary) it is set a large exclusion angle of

120º to avoid the Sun's radiation. Also, it is important to highlight that the Sun's exclusion angle

must not interfere with the field of view of the satellite during all the mission, which means that

none of the observations must be interrupted by the Sun's exclusion angle constraint.

As we did in the previous subsections, we will calculate first the exclusion region, and

then we will compare half cone angles. In this case, we will consider the anti-Sun vector

(𝑟𝑎𝑆/𝑠𝑎𝑡), and the angle of 60º where the satellite can observe the target, 𝛾𝑎𝑆 (see Figure 2.17),

which define a visibility cone. The targets outside this cone will not be visible.

Figure 2.17: Visibility criteria for the constraint of the Sun's exclusion angle, comparison between 𝜽𝒂𝑺 and 𝜸𝒂𝑺, not to scale

To calculate this constraint, we will assume that all the radiation comes from the center

of the Sun, neglecting the radiation that comes from the Sun's surface. We will base this

hypothesis on the method of Aristarchus to measure the relative size and distance of the Sun

and the Earth [12], in order to compare the angle of incidence of the sunlight from the center of

the Sun (𝛿) and from the farthest points of the Sun's surface to the satellite, 𝛼 and 𝛽 (see Figure

2.18).

𝜃𝑎𝑆 𝛾𝑆

𝛾𝑎𝑆

𝑟𝑆/𝑠𝑎𝑡

𝑟𝑎𝑆/𝑠𝑎𝑡

Earth

Sun

𝑅⊕

𝑟𝑠𝑎𝑡/𝐸

𝑟𝑆/𝐸

Sat

Target

𝑟𝐿𝑜𝑆

31

Figure 2.18: Method of Aristarchus to measure the relative size and distance of the Sun, the Earth and the Moon, not to scale

Where 𝑆 is the distance between the Earth and Sun9, 𝑠 is the Sun's radius10, and 𝑡 is

the Earth's radius plus the orbit altitude11

Table 2.2

. Using basic trigonometry we can calculate the three

angles 𝛼, 𝛽, and 𝛿, whose comparison is shown in .

Sunlight angle comparison

δ α β

Angle value (degrees) 2.7 . 10-3 0.264 0.269

Absolute error respect to 𝜹 (degrees)

0.261 0.266

Table 2.2: Deviation angles from considering the sunlight parallel to the Earth's surface or centered on the Sun's nucleus

As we can observe in Table 2.2, the maximum error in case we consider that all the

radiation comes from the center of the Sun is a deviation of less than 0.3 degrees. This error

can be considered in the side of safety if we increase 0.3 degrees the exclusion region.

Therefore, the required visibility half cone angle will be 𝛾𝑎𝑆 = 60° − 0.3° = 59.7°.

Next, we will determine the exclusion region as all the points of the spacecraft centered

celestial sphere outside the visibility cone defined by 𝑟𝑎𝑆/𝑠𝑎𝑡 and 𝛾𝑎𝑆, see Figure 2.19.

9 𝑆 = 1.496 . 108 𝑘𝑚 10 𝑠 = 6.96 . 105 𝑘𝑚 11 Remember that CHEOPS orbit is LEO, so the extra kilometers due to the orbit altitude will not produce a large error in the model considered.

𝛼

𝛽 𝛿

32

Figure 2.19: Spacecraft centered celestial sphere considering the Sun's exclusion angle constraint

Observing Figure 2.19 we can notice that the Sun's exclusion angle constraint is a

strong candidate to be the driver constraint for the visibility calculation. As we can see, the

exclusion region is very large, which makes observable just a few target stars. Also, the visible

region moves very slowly, since the orbit is Sun-synchronous with a period of one year.

Calculating the percentage of visible area respect to all the celestial sphere, we obtain that the

that barely a 17.3% of the sky is visible for each instant of time, considering only the Sun's

exclusion angle constraint.

Next, we will calculate 𝜃𝑎𝑆 similarly to the previous subsections. The Sun's ephemeris

give us the Sun's position vector in the Earth centered J2000 reference frame, 𝑟𝑆/𝐸 [11]. After

that, we will use vector geometry to obtain the anti-Sun's position vector in the satellite centered

reference frame, 𝑟𝑎𝑆/𝑠𝑎𝑡. Then, we will compare 𝜃𝑎𝑆 and 𝛾𝑎𝑆, and we will apply the visibility

criterion to obtain the logical matrix. In this case, the visibility criterion is the opposite to the

considered in the previous subsections, as we are calculating the anti-Sun vector. The visibility

criterion is:

• If 𝜃𝑎𝑆 < 𝛾𝑎𝑆 the target is visible in the time instant considered.

• If 𝜃𝑎𝑆 > 𝛾𝑎𝑆 the target is in the Sun's exclusion region.

-180 -135 -90 -45 0 45 90 135 180

-90

-45

0

45

90

RA (degrees)

dec

(deg

rees

)Spacecraft centered celestial sphere (Sun`s exclusion angle constraint)

-90

-45

0

45

90

-180 -135 -90 -45 0 45 90 135 180

33

Figure 2.20: Exclusion by the Sun time intervals for all the targets and 1.5 months

34

Figure 2.20 represents the exclusion time intervals produced by the Sun's radiation for

1.5 months and all the targets. Next, we will co