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AS3010: Introduction to Space Technology L E C T U R E 3-4 CONTENTS Part B, Lectures 3 & 4 09 March, 2017 A look at what we will cover in this course – orbit representation, orbit determination, orbital manoeuvres; attitude representation, attitude determination, attitude dynamics and control, attitude stabilization; thermal management; communication and telemetry; power systems; Indian space scenario. For this part of the course, we are mainly concerned with satellites and their orbits. The first thing that we are going to look at is ‘how to represent satellite orbits?’ How can two of us identify whether we are talking about the ‘same’ orbit. So we will start with Orbit representation – giving an address to orbits. There are many ways to represent satellite orbits; we will look at one of them – using Keplerian orbital elements. Then we will turn our attention to orbit determination. We talked about orbital transfers in the first part of the course – transfer from current orbit to a desired orbit. But, how do we know our current orbit? That is the topic of Orbit determination. Why orbit determination is needed? Orbit determination is needed exactly because we need to know what is the current orbit of a satellite or a spacecraft. There would have been some uncertainty in the firing of last stage and the satellite would not have been injected into the intended orbit. But, how do we know this? Through orbit determination. We may want to know the orbit of an enemy satellite. Also the orbits of satellites changes over time due to – atmospheric drag in case of low earth orbits, solar radiation pressure, influence of other gravitational bodies, . . . The first successful orbit determination dates back to Gauss. On January 1, 1801, Piazzi, an Italian astronomer, discovered a new celestial object. He thought it was an undocumented star. To make sure whether it is a star or not, he looked for it at the same location at the ‘same’ time, next day. Do stars appear at the same location of sky at same of the day/ night? Yes, if earth rotates exactly 360 degrees in a solar day. However, owing to earth’s revolution about the sun, earth rotates approximately 1 degree more than 360 degrees per day. This brings us to the following definition. Lecture 3 & 4 AS3010: Introduction to Space Technology Definition . Sidereal day Time taken by earth to rotate 360 degrees. It is 23 hours 56 minutes and 4.1 seconds (slightly shorter than a solar day). Sidereal day for earth is shorter than the solar day. Sidereal day for a planet would be longer than its solar day if it were to revolve around the Sun in clockwise direction (when view from North pole down). However, none of the planets do that. Venus has its sidereal day greater than its solar day as it rotates about its axis in clockwise direction unlike most other planets. Piazzi had found Ceres – the first asteroid to be identified. Piazzi first thought it was a comet. Later, experts of the time classified it as a planet (This is because Ceres orbits in the asteroid belt between Mars and Jupiter, and Kepler had predicted, based on his calculations, that there would be a planet there). Ceres, now classified as a dwarf planet, is named after the Roman goddess of growing plants, the harvest, and motherly love. Piazzi observed Ceres for about 41 days, made 22 measurements, and then the object got too close to the Sun’s glare for further observations. Now the question is: where should one look for, in the huge ocean of stars in the sky, to find again what Piazzi had observed? Before that, let us introduce few more concepts. We said Piazzi made 22 measurements. What kind of measurements did Piazzi make? He made some angle measurements – azimuth and elevation. From where should we start measuring azimuth? How can someone else sitting at a different part of earth make use of Piazzi’s measurements to locate the same object that Piazzi spotted? To make this possible, astronomers make their measurements with respect to Celestial coordinate system. Celestial coordinate system is an inertial coordinate system centred at the celestial sphere. Definition . Celestial sphere An imaginary sphere of arbitrary large radius, centred at the observer (centre of earth). All objects in the observer’s sky can be thought of as projected onto the inside surface of this sphere. A star O is represented on a celestial sphere by two angles α (right ascension) and δ (declination). The pair (α, δ) is equivalent to (azimuth, elevation) in the spherical coordinate 2 Lecture 3 & 4 AS3010: Introduction to Space Technology system. For the celestial sphere centred at Earth, X-axis is the vernal equinox vector, and XY plane is the equatorial plane, and Z axis passes through the North Pole. Celestial coordinate system centred at Earth is shown below. In the celestial sphere representation of the sky, the distance information is lost – every object in the sky is identified by its right ascension and declination angles. In the figure above, the star O is identified by (α, δ). (Any other star/ celestial object, which has its centre, in the same direction ahead of or behind O also will be represented by the same point on the celestial sphere.) We said that X-axis is the vernal equinox vector. What is Vernal equinox? The equatorial plane of the earth makes an angle of about 23.4◦ with the ecliptic plane (the plane in which earth revolves around the sun). Now, there is this line of intersection between the equatorial and ecliptic planes. One end of this line points to the star constellation Aries and the other to Libra. This is illustrated in the figure below. Libra Aires Note that the equatorial plane of Earth does not change its orientation while the Earth is rotating or revolving around the Sun. Earth has an angular momentum owing to its rotation about its own axis that is perpendicular to the equatorial plane. As angular momentum is conserved (its direction does not change), the equatorial plane perpendicular to the direction of angular momentum also will not change its direction. Thus, Vernal equinox vector (pointing to Aries) gives a fixed direction in space. The azimuth of a 3 Lecture 3 & 4 AS3010: Introduction to Space Technology celestial object is measured with respect to this fixed direction in space. Typically around March 21 every year the vernal equinox vector from earth passes through the sun, and on that day, the day and the night will have equal lengths – therefore the name equinox! Coming back to the story, what had Piazzi discovered? Was it a planet, a star, a comet, or something else? How to spot it again? None at that time seemed to have a clue about it except for Gauss who was then 24 years old. Gauss used 3 measurements of Piazzi along with the Keplarian laws of orbital motion to determine the orbit of Ceres. About an year later after its first sighting, Ceres was found again very close to where Gauss predicted it would be at that time! On January 1, 1802, Ceres was found within 0.5 degrees of where Gauss predicted it would be (Ceres could have been anywhere in the sky). To have an idea about accuracy, 0.5 degrees is the angle subtended by full moon. Let us make precise, the notion of angle subtended by a celestial object. Definition . Angular diameter (γ) Angle a celestial object subtends at the point of observation. Coincidently, the sun and the moon have approximately same angular diameters: γsun = 0.533 degrees and γmoon = 0.515 degrees. A related notion is that of solid angle. Definition . Solid angle (Ω) Solid angle is the area of the segment of a unit sphere that an object covers. It is measured in steradians (sr). The solid angle of an area on the surface of a unit sphere is the area itself. For a sphere of radius r, the solid angle subtended by an area A on its surface is given as Ω= 4 A r2 Lecture 3 & 4 AS3010: Introduction to Space Technology Thus, the solid angle of sphere viewed from its center is Ω = 4π sr. Clearly, the solid angle of a face of cube viewed from its center should be one sixth of 4π 2 and is equal to Ω = 4π/6 = π sr. 3 In general, solid angle of a surface S is ZZ ~r · n̂ Ω= ds r|3 S |~ If this surface forms part of a sphere, then ZZ sin θdθ dφ Ω= S This is clear from the figure below. 5 Lecture 3 & 4 AS3010: Introduction to Space Technology Exercise . Show that a cone of apex angle 2θ has a solid angle of 2π(1 − cos θ). Recall that we said, we are going to look at the topics of orbit representation first, and then orbit determination. We discussed the history of orbit determination of Ceres by the great mathematician Gauss. A description of the principles used by Gauss for the orbital determination of Ceres can be found at http://www.keplersdiscovery.com/Asteroid.html Gauss used just 3 measurements to make an initial estimate of the orbit. Three measurements would have been enough if the measurements were exact. However, all real world measurements are noisy. Gauss used rest of the measurements to ‘correct’ the initially estimated trajectory. The history was just for motivation – we will not analyse what Gauss did. Rather, we would take up a much simpler problem and try to understand the basic principles behind orbit determination. Orbit determination is all about determining the orbit of a celestial object using measurements of its position and velocity. A good reference for the topic of orbit determination is Prussing, J. E. and Conway, B. A., Orbital Mechanics, Oxford University Press, 1993. One of the uses of orbit determination is for orbit correction as we said before. Another use would be to determine the orbit of an ‘enemy’ satellite or a comet or any other celestial body. The only measurement that Piazzi could have made is the angular position of the celestial object. Thus, only this information was available to Gauss for orbit determination. However, for a satellite, range, range rate, altitude above ground, position in three dimensions (using GPS), etc. forms some of the measurements that can be made today. Modern measurement techniques involve the use of radar, laser, . . . which gives range and range rate information. Currently, GPS is used for accurate orbit determination of satellites. Orbit determination algorithms these days can take advantage of these – a luxury that Gauss did not have! We will look at what are the current orbital measurement techniques, also known as satellite tracking. We will also look at the mathematical technique of least square estimation that is used for orbit determination in the presence of noisy measurements. Till now we have looked at satellite or a spacecraft as a point in its orbit. However, a satellite is a rigid body with finite size. A satellite (in general, any rigid body) has position and orientation in space. Position of the satellite is determined by its orbit – we have already studied orbital mechanics. The orientation of a satellite in space is called its attitude. What we will look at, in most part of rest of the course, is how to determine and control a satellite’s attitude. In doing this, we will also study the attitude dynamics of a satellite (the theory is valid for attitude dynamics of any rigid body, and therefore useful for flight dynamics too). 6 Lecture 3 & 4 AS3010: Introduction to Space Technology As said above, when we consider satellite as a rigid body, apart from position, it will also have an orientation or attitude. Then the question is ‘How to represent a satellite’s orientation?’ This takes us to the topic of attitude representation. Attitude representation Every satellite has a specific mission. If it is a telescopic satellite, then it needs to ‘point to’ the relevant celestial object (a star, sun, some distant galaxy, . . . ). If it is a remote sensing satellite or a communication satellite, then it needs to point down at some location on earth. In any case, the orientation or the attitude of a satellite is important. Before controlling orientation, we need to understand what do we mean by orientation of a satellite. Therefore, we will first look at how to represent the orientation/ attitude of a satellite (or any rigid body) in space – that is, with respect to an inertial frame. There are many ways to do this. You would have already studied the Euler angle representation of attitude of a rigid body in your Flight Dynamics II course. Although Euler angles can be used to represent satellite orientation, something called as quaternion representation is more popular among the satellite community. We will try to look at what quaternions are and how they help in attitude representation and attitude computations. Once we know how to represent attitude, the next question that bothers us is “How do we ‘measure’ this attitude?” Thus, it becomes the next topic that we will dwell on – attitude determination. Attitude determination As said earlier, a satellite is required to orient in space in a desired direction. To make a satellite point in a desired direction, it is first necessary to determine/ estimate what is the current orientation of the satellite. And, if the current orientation is not the desired one, then we make corrections. To determine the orientation, we need references. What are the reference objects out there for the satellite to look at? Of course, other celestial bodies – earth (largest object in the satellite’s sky), sun (brightest object), stars (being ‘at infinity’ they are fixed objects in the sky), . . . A satellite need to look at these objects as references for it to determine its attitude. This is typically achieved using sensors. A satellite will have internal reference and external reference sensors. Internal sensors include gyros and rate gyros. We will look at different types of gyros and the principles based on which they operate. External sensors measures the angle between the satellite and some external reference. Depending on which external object is used as a reference, we have sun sensors, star sensors, earth/horizon sensors . . . We will try to understand how a satellite’s orientation can be determined using these sensors. 7 Lecture 3 & 4 AS3010: Introduction to Space Technology Once the attitude is determined, a satellite needs to adjust/ control the attitude to comply to its mission requirements. Doing this effectively requires a good understanding of Attitude/ Rotational Dynamics. Attitude dynamics After determining the current attitude, the satellite needs to be re-oriented to the desired attitude. To do this appropriately, we need to have a good understanding of the satellite attitude dynamics. We will study the rotational dynamics of the satellite and analyse the resulting system of equations and its solutions. This will be more or less the kind of dynamics that you already studied in your high school or in your engineering mechanics course. A good understanding of attitude dynamics will equip us to look at techniques for attitude control of a satellite. Attitude control As said earlier, attitude control is required to point a satellite in the desired direction. Also, it may be required to orient the main thruster before firing it for orbital transfer manoeuvres. Thus, orienting the satellite in particular desired directions is very important for a satellite’s mission. The satellite attitude is controlled using actuators. The actuators used for attitude control in satellites include reaction wheels, control moment gyros, and thrusters. We will discuss working principles of some of these. Is it enough to control/ adjust a satellite’s attitude once, or is it necessary to correct it periodically? A periodic correction is required if disturbance torques take the satellite away from required attitude. The typical disturbance torques a satellite experiences are • atmospheric drag • solar radiation pressure • gravity gradient • magnetic torque .. . Instead of constantly correcting the attitude, it is desirable to ‘stabilize’ the attitude. We will briefly discuss a couple of attitude stabilization techniques. 8 Lecture 3 & 4 AS3010: Introduction to Space Technology Attitude stabilization After determining the attitude, if the satellite is found to be not pointing in the ‘right’ direction, it needs to be re-oriented. Rather than actively controlling the attitude time and again, it is desirable to stabilize the attitude in some fashion. A technique for passive stabilization is to make use of earth’s gravity gradient. Earth’s gravitational force varies with distance. As a result, a body that is closer to earth gets attracted more than that which is away. Thus a body with ‘sufficient length’ tends to align towards earth, as parts of it ‘nearer’ to earth get attracted more. A satellite can exploit this property to point towards earth. In the above figure, for a satellite that is like a dumb-bell, point B that is closer to earth gets attracted more than point A resulting in a torque that aligns satellite along earth’s centre. Another passive stabilization technique would be to give the satellite a spin about an axis. Then, we expect that the conservation of angular momentum will help keep the attitude of the satellite (like a spinning top not falling). This strategy of stabilization via giving the satellite an spin was employed to stabilize Explorer I, the first US satellite launched in January 31, 1958. (Explorer I looks like a long slender rod. The launch of Explorer I was a response to the Soviet satellite Sputnik I that went into orbit on October 4, 1957. Sputnik I was spherical in shape. Does that explain the geometry of Explorer I?) Explorer I was intended to be stabilized about its minor axis (axis with least moment of 9 Lecture 3 & 4 AS3010: Introduction to Space Technology inertia), and therefore an angular momentum was given about that axis while it was put in the orbit. However, within some time of deployment, Explorer I went into a flat spin – a spin about its major axis. Why did this happen? We may need to understand more of rotational/ attitude dynamics before we answer this question. Thermal management In the beginning of the movie ‘Gravity’ by Alfonso Cuaron that appeared in 2013, the following words appear in the screen at the very beginning of the movie: “At 600 km above planet Earth, the temperature fluctuates between +258 and -148 degrees Fahrenheit. There is nothing to carry sound – no air pressure, no oxygen. Life in space is impossible . . . ” With apprehensions about the technical accuracy of these numbers, let us ask the question ‘Why does the temperature fluctuate out there in space?’. That side of the satellite facing the sun gets heated up due to solar radiation. Also, when the satellite is in the shade of earth, whole of it ‘freezes’. Thermal management/ control in satellite is extremely important as some of the payloads have electronic components that are extremely sensitive to temperature, and are required to operate within a specified temperature range. The thermal control system maintains the temperature of equipments within the desired limits. Both active and passive cooling techniques are used. Typical components used are radiators, louvres, coatings, heaters, . . . We will look at some of the temperature control aspects in the satellite. 10 Lecture 3 & 4 AS3010: Introduction to Space Technology Communication and telemetry It is important for the satellite to communicate with a base station to receive commands and transmit data that it collected, for example, in the case of a remote sensing satellite. We will have a brief look into these aspects; get us familiarised with terms like S-band, Ku-band, . . . that we hear often about. Power systems Another important sub-system of a satellite is its power system. Initial satellites used (non-rechargeable) batteries, and the satellite was ‘dead’ when the batteries were dead. Life of the power system thus determines the life of a satellite. Thus one should look at ways to generate power on-board. Sun is the obvious source of energy for satellite. Most satellites these days work on solar cells that powers the satellite sub-systems and charges batteries that will back-up the systems when the satellite is in earth’s shade. Apart from solar power, nuclear power, fuel cells, . . . are the other power sources that are currently used in satellites. Electrical power systems generates, stores, conditions, control, and distribute power within the specified voltage/ current levels to all the sub-systems. We will have a quick peep at these and the design considerations for satellite power systems. Space technology: The Indian scenario The space research in India was initiated in 1962, not much later than the first satellite Sputnik 1 was launched, under the visionary Dr. Vikram Sarabhai. This timely start of the national space programme, gave us today a respectable position among the global space community – self-sufficient to launch our own satellites and indulge in space exploratory missions. We will have a quick look at the history of evolution of Indian space technology, and review where it stands today. Text books/ References Various portions of the syllabus would be covered from the following books/ references: 1. Kaplan, M. H., Modern Spacecraft Dynamics & Control, John Wiley & Sons, 1976. 2. Wertz, J. R., Spacecraft Attitude Determination and Control,, Kluwer Academic Publishers, 1978. 11 Lecture 3 & 4 AS3010: Introduction to Space Technology 3. Sidi, M. J., Spacecraft Dynamics & Control: A Practical Engineering Approach, Cambridge University Press, 1997. 4. Course notes on Spacecraft Attitude Dynamics and Control by Prof. Franco Bernelli Zazzera of Politecnico di Milano, Italy. (available online) Evaluation This part of the course has a total of 50 marks, and it will be distributed (tentatively) as follows: • Quiz 2: 20 marks • End semester exam: 30 marks Exam will involve mostly theory questions – definitions, facts, . . . and some numerical problems, of course. If you have sat through every lecture and took down notes seriously, then I do not see you not getting 100% marks for this part of the course. Attendance requirement for this part of the course is 85% – you can bunk at the maximum 8 classes (whatever the reasons be); that already is too much allowance, and anything more than that – you are getting a ‘W’ for sure. 12