AS3010: Introduction to Space Technology

L E C T U R E 3-4

Part B, Lectures 3 & 4

09 March, 2017

C O N T E N T S

A look at what we will cover in this course – orbit representa-tion, orbit determination, orbital manoeuvres; attitude repre-sentation, attitude determination, attitude dynamics and con-trol, attitude stabilization; thermal management; communica-tion and telemetry; power systems; Indian space scenario.

For this part of the course, we are mainly concerned with satellites and their orbits. Thefirst thing that we are going to look at is ‘how to represent satellite orbits?’ How can twoof us identify whether we are talking about the ‘same’ orbit. So we will start with Orbitrepresentation – giving an address to orbits. There are many ways to represent satelliteorbits; we will look at one of them – using Keplerian orbital elements. Then we will turnour attention to orbit determination. We talked about orbital transfers in the first partof the course – transfer from current orbit to a desired orbit. But, how do we know ourcurrent orbit? That is the topic of Orbit determination.

Why orbit determination is needed? Orbit determination is needed exactly because weneed to know what is the current orbit of a satellite or a spacecraft. There would havebeen some uncertainty in the firing of last stage and the satellite would not have beeninjected 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 changesover 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 undocu-mented 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’srevolution about the sun, earth rotates approximately 1 degree more than 360 degrees perday. This brings us to the following definition.

AS3010: Introduction to Space Technology Lecture 3 & 4

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 belonger than its solar day if it were to revolve around the Sun in clockwise direction (whenview from North pole down). However, none of the planets do that. Venus has its siderealday greater than its solar day as it rotates about its axis in clockwise direction unlikemost other planets.

Piazzi had found Ceres – the first asteroid to be identified. Piazzi first thought it was acomet. Later, experts of the time classified it as a planet (This is because Ceres orbitsin the asteroid belt between Mars and Jupiter, and Kepler had predicted, based on hiscalculations, 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 gottoo 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, tofind 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 – az-imuth and elevation. From where should we start measuring azimuth? How can someoneelse sitting at a different part of earth make use of Piazzi’s measurements to locate thesame object that Piazzi spotted?

To make this possible, astronomers make their measurements with respect to Celestialcoordinate system. Celestial coordinate system is an inertial coordinate system centredat 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 ofthis sphere.

A star O is represented on a celestial sphere by two angles α (right ascension) and δ (dec-lination). The pair (α, δ) is equivalent to (azimuth, elevation) in the spherical coordinate

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AS3010: Introduction to Space Technology Lecture 3 & 4

system.

For the celestial sphere centred at Earth, X-axis is the vernal equinox vector, and X-Y plane is the equatorial plane, and Z axis passes through the North Pole. Celestialcoordinate system centred at Earth is shown below.

In the celestial sphere representation of the sky, the distance information is lost – everyobject in the sky is identified by its right ascension and declination angles. In the figureabove, the star O is identified by (α, δ). (Any other star/ celestial object, which has itscentre, in the same direction ahead of or behind O also will be represented by the samepoint on the celestial sphere.)

We said that X-axis is the vernal equinox vector. What is Vernal equinox? The equatorialplane of the earth makes an angle of about 23.4 with the ecliptic plane (the plane inwhich earth revolves around the sun). Now, there is this line of intersection between theequatorial and ecliptic planes. One end of this line points to the star constellation Ariesand the other to Libra. This is illustrated in the figure below.

Aires

Libra

Note that the equatorial plane of Earth does not change its orientation while the Earthis rotating or revolving around the Sun. Earth has an angular momentum owing toits rotation about its own axis that is perpendicular to the equatorial plane. As angularmomentum is conserved (its direction does not change), the equatorial plane perpendicularto the direction of angular momentum also will not change its direction. Thus, Vernalequinox vector (pointing to Aries) gives a fixed direction in space. The azimuth of a

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AS3010: Introduction to Space Technology Lecture 3 & 4

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 throughthe sun, and on that day, the day and the night will have equal lengths – therefore thename 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 aboutit except for Gauss who was then 24 years old.

Gauss used 3 measurements of Piazzi along with the Keplarian laws of orbital motionto determine the orbit of Ceres. About an year later after its first sighting, Ceres wasfound 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 itwould 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 anglesubtended 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 measuredin steradians (sr).

The solid angle of an area on the surface of a unit sphere is the area itself. For a sphereof radius r, the solid angle subtended by an area A on its surface is given as

Ω =A

r2

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AS3010: Introduction to Space Technology Lecture 3 & 4

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

and is equal to Ω = 4π/6 =2

3π sr.

In general, solid angle of a surface S is

Ω =

∫∫S

~r · n|~r|3

ds

If this surface forms part of a sphere, then

Ω =

∫∫S

sin θdθ dφ

This is clear from the figure below.

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AS3010: Introduction to Space Technology Lecture 3 & 4

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, andthen orbit determination. We discussed the history of orbit determination of Ceres by thegreat mathematician Gauss. A description of the principles used by Gauss for the orbitaldetermination 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 measure-ments would have been enough if the measurements were exact. However, all real worldmeasurements are noisy. Gauss used rest of the measurements to ‘correct’ the initiallyestimated trajectory.

The history was just for motivation – we will not analyse what Gauss did. Rather, wewould take up a much simpler problem and try to understand the basic principles behindorbit determination. Orbit determination is all about determining the orbit of a celestialobject 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. Anotheruse would be to determine the orbit of an ‘enemy’ satellite or a comet or any other celestialbody.

The only measurement that Piazzi could have made is the angular position of the celestialobject. Thus, only this information was available to Gauss for orbit determination. How-ever, 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. Modernmeasurement techniques involve the use of radar, laser, . . . which gives range and rangerate information. Currently, GPS is used for accurate orbit determination of satellites.Orbit determination algorithms these days can take advantage of these – a luxury thatGauss 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 usedfor 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) hasposition and orientation in space. Position of the satellite is determined by its orbit – wehave already studied orbital mechanics. The orientation of a satellite in space is calledits attitude. What we will look at, in most part of rest of the course, is how to determineand control a satellite’s attitude. In doing this, we will also study the attitude dynamicsof a satellite (the theory is valid for attitude dynamics of any rigid body, and thereforeuseful for flight dynamics too).

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AS3010: Introduction to Space Technology Lecture 3 & 4

As said above, when we consider satellite as a rigid body, apart from position, it willalso have an orientation or attitude. Then the question is ‘How to represent a satellite’sorientation?’ 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 ‘pointto’ the relevant celestial object (a star, sun, some distant galaxy, . . . ). If it is a remotesensing satellite or a communication satellite, then it needs to point down at some locationon 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 ofa satellite. Therefore, we will first look at how to represent the orientation/ attitude of asatellite (or any rigid body) in space – that is, with respect to an inertial frame. Thereare many ways to do this. You would have already studied the Euler angle representationof attitude of a rigid body in your Flight Dynamics II course. Although Euler angles canbe used to represent satellite orientation, something called as quaternion representationis more popular among the satellite community. We will try to look at what quaternionsare 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 – attitudedetermination.

Attitude determination

As said earlier, a satellite is required to orient in space in a desired direction. To make asatellite point in a desired direction, it is first necessary to determine/ estimate what isthe current orientation of the satellite. And, if the current orientation is not the desiredone, then we make corrections.

To determine the orientation, we need references. What are the reference objects out therefor the satellite to look at? Of course, other celestial bodies – earth (largest object in thesatellite’s sky), sun (brightest object), stars (being ‘at infinity’ they are fixed objects inthe sky), . . .

A satellite need to look at these objects as references for it to determine its attitude. Thisis typically achieved using sensors. A satellite will have internal reference and externalreference sensors.

Internal sensors include gyros and rate gyros. We will look at different types of gyros andthe principles based on which they operate. External sensors measures the angle betweenthe satellite and some external reference. Depending on which external object is used asa reference, we have sun sensors, star sensors, earth/horizon sensors . . . We will try tounderstand how a satellite’s orientation can be determined using these sensors.

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AS3010: Introduction to Space Technology Lecture 3 & 4

Once the attitude is determined, a satellite needs to adjust/ control the attitude to com-ply to its mission requirements. Doing this effectively requires a good understanding ofAttitude/ Rotational Dynamics.

Attitude dynamics

After determining the current attitude, the satellite needs to be re-oriented to the desiredattitude. To do this appropriately, we need to have a good understanding of the satelliteattitude dynamics. We will study the rotational dynamics of the satellite and analysethe resulting system of equations and its solutions. This will be more or less the kind ofdynamics that you already studied in your high school or in your engineering mechanicscourse. A good understanding of attitude dynamics will equip us to look at techniquesfor 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 transfermanoeuvres. Thus, orienting the satellite in particular desired directions is very importantfor a satellite’s mission.

The satellite attitude is controlled using actuators. The actuators used for attitude controlin satellites include reaction wheels, control moment gyros, and thrusters. We will discussworking principles of some of these.

Is it enough to control/ adjust a satellite’s attitude once, or is it necessary to correct itperiodically? A periodic correction is required if disturbance torques take the satelliteaway 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. Wewill briefly discuss a couple of attitude stabilization techniques.

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AS3010: Introduction to Space Technology Lecture 3 & 4

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 timeand 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’sgravitational force varies with distance. As a result, a body that is closer to earth getsattracted more than that which is away. Thus a body with ‘sufficient length’ tends toalign towards earth, as parts of it ‘nearer’ to earth get attracted more. A satellite canexploit 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 earthgets attracted more than point A resulting in a torque that aligns satellite along earth’scentre.

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 attitudeof the satellite (like a spinning top not falling).

This strategy of stabilization via giving the satellite an spin was employed to stabilizeExplorer I, the first US satellite launched in January 31, 1958. (Explorer I looks like along slender rod. The launch of Explorer I was a response to the Soviet satellite Sputnik Ithat went into orbit on October 4, 1957. Sputnik I was spherical in shape. Does thatexplain the geometry of Explorer I?)

Explorer I was intended to be stabilized about its minor axis (axis with least moment of

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AS3010: Introduction to Space Technology Lecture 3 & 4

inertia), and therefore an angular momentum was given about that axis while it was putin 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 dynamicsbefore we answer this question.

Thermal management

In the beginning of the movie ‘Gravity’ by Alfonso Cuaron that appeared in 2013, thefollowing 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 degreesFahrenheit. There is nothing to carry sound – no air pressure, no oxygen. Life in spaceis 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 facingthe sun gets heated up due to solar radiation. Also, when the satellite is in the shadeof earth, whole of it ‘freezes’. Thermal management/ control in satellite is extremelyimportant as some of the payloads have electronic components that are extremely sensitiveto temperature, and are required to operate within a specified temperature range.

The thermal control system maintains the temperature of equipments within the desiredlimits. Both active and passive cooling techniques are used. Typical components used areradiators, louvres, coatings, heaters, . . .

We will look at some of the temperature control aspects in the satellite.

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AS3010: Introduction to Space Technology Lecture 3 & 4

Communication and telemetry

It is important for the satellite to communicate with a base station to receive commandsand 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 atways to generate power on-board. Sun is the obvious source of energy for satellite. Mostsatellites these days work on solar cells that powers the satellite sub-systems and chargesbatteries 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 arecurrently used in satellites.

Electrical power systems generates, stores, conditions, control, and distribute powerwithin the specified voltage/ current levels to all the sub-systems.

We will have a quick peep at these and the design considerations for satellite powersystems.

Space technology: The Indian scenario

The space research in India was initiated in 1962, not much later than the first satelliteSputnik 1 was launched, under the visionary Dr. Vikram Sarabhai. This timely start of thenational space programme, gave us today a respectable position among the global spacecommunity – self-sufficient to launch our own satellites and indulge in space exploratorymissions. 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 AcademicPublishers, 1978.

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AS3010: Introduction to Space Technology Lecture 3 & 4

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 BernelliZazzera 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) asfollows:

• Quiz 2: 20 marks

• End semester exam: 30 marks

Exam will involve mostly theory questions – definitions, facts, . . . and some numericalproblems, 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 maximum8 classes (whatever the reasons be); that already is too much allowance, and anythingmore than that – you are getting a ‘W’ for sure.

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