AS3010: Introduction to Space Technology

Transcription

1 AS3010: Introduction to Space Technology L E C T U R E 6 Part B, Lecture 6 17 March, 2017 C O N T E N T S In this lecture, we will look at various existing satellite tracking techniques. Recall that we were looking at the problem of Orbit representation. We saw that there is no unique way to do this. We chose to represent a satellite s orbit using Keplerian orbital elements. This representation works for any body that has elliptic orbits. Recall that the Keplerain orbital elements included the size and shape of orbits given by semi-major axis (a) and eccentricity (e). Then it specified where on the orbit the satellite is by specifying anomaly. The saw the concepts of true anomaly θ eccentric anomaly E and mean anomaly M defined as M = 2π t T

2 where t is the time since the last passage of perigee, and T is the time period. The mean anomaly and the eccentric anomaly are related through Kepler s equation given as M = E e sin E Then, we had to orient the orbit around Earth for which we used 3 angles (Ω, i, ω) where Ω is the longitude of ascending node, i is the inclination, and ω is called the argument of perigee. A pictorial representation of this is given below. The angles (Ω, i, ω) can be looked upon as Euler angles. We will see how to use three rotations to orient the orbit with respect to an inertial coordinate system, in our case, the celestial coordinate system centered at Earth. We start with the orbit with a set of axis attached to it that rotates along with it (body axis) which is aligned along the axis of the celestial coordinate system. This is shown below. Now we perform our 3 rotations. (1) Rotate by Ω about Z axis. 2

3 By the end of first rotation, we have the following situation (2) Rotate by i about X axis. Here the X axis the body X axis. The situation before and after this rotation as looked at from in front of X axis is illustrated below. By the end of this rotation, the line of apsides the line connecting the perigee and apogee are on the equatorial plane. Perigee is the ascending node and apogee is the descending node. The angle or argument the perigee makes with the line of nodes, measured from the ascending node, is the final rotation. (3) Rotate by ω about Z axis. The above three Euler rotations, (Ω, i, ω) about Z X Z, orients the orbit about Earth. Like any Euler angle representation, this representation also suffers from singularity. Euler angle rotations to orient a body with respect to an inertial axis works if the rotations are about 3 different axes when viewed from the inertial frame. This is not the case when i = 0 that is, the second rotation rotation about X axis is zero. In this case, the first and the last body Z axis are along the same inertial direction, and as a result, one will not be able to distinguish between the first and the third rotation angles. In other words, when i = 0, we cannot distinguish between Ω and ω. Any set of angles (Ω, 0, ω) such that Ω+ω is a given constant will represent the same final orbit. This makes the inverse mapping ill-defined leading to singularity. This is depicted for a particular case below. 3

4 We said the the six parameters (a, e, θ, Ω, i, ω) completely determines the orbit of a satellite. But, how do we find these parameters for a given satellite. This is the question of orbit determination. Recall how Gauss did this he used measurements from Piazzi to estimate the orbit. That is what orbital determination is all about estimating orbital elements from measurements. In the case of Piazzi, the measurements were only angular measurements of positions only. However, for a satellite, we can get additional measurements like range, range rate, absolute positions and velocities,... The process of making measurements of position and velocity of a satellite is called Satellite tracking. We will now look at a few satellite tracking techniques. 1) Optical tracking Optical tracking typically involves angular measurements (azimuth and elevation). Typical instruments used are cameras, and optical tracking usually provides with no distance information. 2) Radio tracking Radio tracking is also called Doppler tracking, as Doppler effect is made use of in it. It is based on the difference between transmitted (by satellite) and received (by a receiver on ground) frequencies. Because, the satellite and the receiver have a relative velocity, the transmitted and the observed frequencies will be different due to Doppler effect. The scenario is illustrated below. The observed frequency is given as f obs = f source 1 Vr c (1) 4

5 where f obs is received frequency, f source is transmitted frequency, c is the speed of light, and V r is component of V along the line of transmission that is, V r = V cos θ, where θ is the angle that the velocity vector makes with the line of transmission. The receiver tower records the observed frequency history as the satellite passes over the receiver tower. Such a plot of the time history of the observed frequency is shown below. From the figure and Equation 1, it is clear that the observed frequency is equal to the transmission frequency when the satellite is directly above the receiver as V r = 0 in this case. Assuming that the path followed by the satellite is approximately a straight line (which may be true for a satellites with nearest point of approach very close to earth s surface?), the asymptotic values of the observed frequencies gives, from Equation 1, an estimate of the velocity of the satellite. Other assumptions made in this method include V r c, that is, no relativistic effects. Also, the earth s rotation is neglected. The angle a satellite makes with the local horizontal (θ) can be measured using multiple receivers separated by known distances. A sample scenario with two receivers is shown in the figure below. Let λ be the wavelength, D the distance between the receivers, and d the extra distance the signal has to travel to reach the second receiver. These can be expressed in terms of 5

6 λ as d = nλ D = mλ where, m is known as D is known, and n can be computed from the interference between signals received at both the receivers. From the knowledge of m and n, the angle the position of the satellite makes with the horizontal is given as cos θ = n m The phase difference between signals at the two receivers is given as φ + 2πk, k = 0, 1, 2,... An additional distance of λ travelled by the signal corresponds to a phase difference of 2π. Thus the additional distance d travelled corresponds to phase difference 2πd λ. This implies that the phase difference is 2πd λ from which, we get n as 2πn = φ + 2πk n = φ 2π or φ 2π + 1 or φ 2π = 2πn (using d = nλ). Thus, we have Once m and n are known, the angle the location of the satellite makes with the local horizontal θ can be computed. The ambiguity in n as above can result in multiple solutions for θ. However, this can easily be overcome by the use of multiple receivers separated by different distances. 3) Radar tracking Radar sends out sharp short signals, and receives reflected signals. From this, the range and direction of the object that reflected the signal can be computed. In general, radar does not give range rate. However, range rate can be obtained using a Doppler radar which uses similar methods as mentioned above to compute relative velocity. 4) Laser ranging Laser ranging is possible only for satellites that are equipped with reflectors. Operational principle is same as that of radar. Laser ranging depends on weather tracking using laser may not be possible, for example, if it is cloudy. 6

7 5) Global Navigation Satellite System (GNSS) Global Navigation Satellite System (GNSS) uses the principle of triangulation if you know your relative distance from certain sources who accurately know their own absolute positions, then your absolute position can also be determined with high accuracy. Various space agencies have their own satellite network providing GNSS. Examples are GPS (USA), GLONASS (Russia), Galileo (European Space Agency), Beidou (China). Apart from global navigation systems, there are regional navigation systems and augmented navigation systems that is, information from global navigation systems augmented using data from regional satellite network. Indian Regional Navigation Satellite System also know as NAVIC is India s regional satellite navigation system. GAGAN: GPS Aided GEO Augmented Navigation of ISRO is an augmented navigation system. GNSS can be used to determine the position and velocity of a satellite. satellite should have appropriate GNSS receiver. For this the Exercise What are the advantages/ disadvantages/ limitations of each of the above satellite tracking methods? ISRO has its satellite tracking center in Bangalore, and it is called ISTRAC - ISRO Telemetry, Tracking and Command Network. Exercise An alien spacecraft, flying past close to Earth, emits a constant frequency signal that is picked up by a receiver on Earth as illustrated in the figure below. f observed Spacecraft v f source v r Receiver f source Earth f source The observed frequency with time, as the spacecraft flew past the receiver, is also shown in the figure. The plot is such that at t = 0, the spacecraft is directly above the receiver. Assumptions to be made: (i) the trajectory of the spacecraft is a straight line, (ii) the velocity of the spacecraft is constant, and (iii) the speed of light is m/s. Additional facts: (i) f source is 1 GHz, and (ii) the slope of the observed frequency curve at t = 0 is 27 GHz per second. Find, 1. velocity of the alien spacecraft, and 2. its distance of closest approach to Earth. t 7

8 8