Satellite meteorology

Transcription

1 GPHS 422 Satellite meteorology GPHS 422 Satellite meteorology Lecture 1 6 July 2012

2 Course outline

3 Course outline continued 10:00 to 12:00 3

4 Course outline continued 4

5 Some reading 5

6 Other books - orbital dynamics 6

7 Other books - orbital dynamics 7

8 Course website 8

9 Course website - continued 9

10 Course website - continued 10

11 Course website - continued 11

12 Satellite orbits An understanding of satellite orbits is crucial to the understanding of how the various meteorological satellites operate. In this course we will look at two main types of orbit used by operational meteorological satellites. These are:- Polar orbiting satellites which are in near-circular sunsynchronous orbits Geostationary satellites We will also briefly discuss other types of satellite orbit, such as the highly elliptical Molniya orbit, and speculate on its suitability in future operational meteorological satellite programmes. 12

13 Newton s Laws of Motion Newton s laws of motion are as follows:- Every body will continue in its state of rest or of uniform motion in a straight line unless acted upon by an impressed force. The rate of change of momentum is proportional to the impressed force and takes place in the line in which the force acts. Action and reaction are equal and opposite. Newton s Second Law is often expressed as:- F = d(mv) dt = m dv dt + v dm dt where F is the force, m is the mass, a is the acceleration, and v is the velocity. For the situation where mass is a constant, equation (1) can be written in the more familiar form:- (1) F = m dv dt (2) 13

14 Newton s Law of Universal Gravitation Newton s Law of Universal Gravitation tells us that the attraction force, F, between two point masses m 1 and m 2 separated by a distance r is F = Gm 1m 2 r 2 (3) where G, is the universal constant of gravitation, G = Nm 2 kg 1. 14

15 Circular orbits For a body in circular motion the centripetal force required is F = mv2 r (4) where v is the orbital velocity of the satellite. The fact that a satellite stays in an orbit around the earth is due to the balance that occurs between the gravitational force and the centripetal force. This balance is also the reason why astronauts and cosmonauts are in a weightless state in space. If we equate the two forces in (3) and (4) we can write mv 2 r = Gm em r 2 (5) where m e, is the mass of the Earth, m e = kg. The mass m, of the satellite can be eliminated from the equation (5). In other words the orbit of the satellite is independent of its mass. 15

16 of its mass. Circular orbits - continued Figure 1 A circular satellite orbit The circumference of the orbit is 2πr and if we divide this by the velocity v, we obtain an expression for the period T, of the orbit T = 2πr v (6) substituting for v in equation (5) gives T 2 = 4π2 Gm e r 3 (7) 16

17 Low earth orbit - example Let us consider a satellite in a circular orbit 850 km above the Earth s surface. The equatorial radius of the Earth is 6378 km. Therefore we can substitute this value to obtain an estimate of the period of these satellites. T = 4π 2 Gm e r 3 = minutes The NOAA polar orbiters have similar orbital characteristics to this, we will discuss NOAA satellites in detail later. 17

18 Example - geosynchronous satellites We can calculate the height of the orbit of Geosynchronous satellites by rearranging equation (7) to give r = 3 T 2 Gm e 4π 2 (8) The Earth completes one revolution in 1 sidereal day which equals 23 hours 56 minutes 4.1 seconds or 86,164.1 seconds (NOTE not 24 hours). If we use the angular velocity ξ, of the Earth, r is then given as Gm r = 3 e ξ 2 (9) ξ is equal to s 1, so substituting in equation (9) r = ( ) 2 = km Therefore the satellite is = km above the Earth s surface. Later we will see that this simple calculation gives a very good estimate of the actual altitude of the GMS-5 satellite. 18

19 Elliptical orbits - Kepler s Laws Most meteorological satellites are launched into circular or near circular orbits. However it is appropriate to consider the more general case of elliptical satellite orbits. Kepler s Laws were formulated, nearly 400 years ago, to explain planetary motion around the sun. These empirical laws were based on the detailed observations of planetary motion by earlier astronomers, particularly Tyco Brahe. The laws are as follows:- All planets travel in elliptical paths with the sun at one focus. The radius vector from the sun to the planet sweeps out equal areas in equal times. (We have already discussed this in relation to earth s elliptical motion around the sun in GPHS422 and the consequences of shorter but more intense Southern Hemisphere summers) The ratio of the square of the period of revolution of a planet to the cube of its semimajor axis is the same for all planets revolving around the sun. 19

20 Elliptical satellite orbit Kepler s Laws are equally valid for describing motion of satellites around the Earth. Figure 2 An elliptical satellite orbit Figure 2 shows the geometry of an elliptical orbit with the Earth at one foci of the ellipse. The closest approach of the orbit to the Earth is called the perigee and the furthest point is refered to as the apogee of the orbit. The apogee is π radians away from the perigee. 20

21 The distance from the centre of the ellipse to either apogee or perigee is the semi-major axis, a of the ellipse and the distance from the focus of the ellipse to the centre of the ellipse, a ɛ divided by the semi-major axis is called the eccentricity ɛ. The eccentricty ranges in value (0 ɛ < 1). For a circle ɛ = 0. The path that the satellite follows is given by r = a(1 ɛ2 ) 1 + ɛ cos θ (10) The angle θ given in equation (10) is the true anomaly and is measured counterclockwise from the perigee. From Kepler s Second Law it is apparent that when the satellite is nearer to perigee its angular velocity will be greater than when it is nearer apogee. The position of the satellite as a function of time is given by Kepler s equation: M = n(t t p ) = e ɛ sin e (11) where M is the mean anomaly; M increases linearly in time at the rate n, called the mean motion constant, given by n = 2π T = Gme a 3 (12) M is zero when the satellite is at perigee, thus t p is the time of perigeal passage. 21

22 Relationship between true and eccentric anomaly Figure 3 Relationship between true anomaly θ and eccentric anomaly e The angle e, shown in figure 3 is the eccentric anomaly. It is related geometrically to the true anomaly: cos θ = cos e ɛ (13) 1 ɛ cos e and cos e = cos θ + ɛ 1 + ɛ cos θ (14) 22

23 An inertial coordinate system By calculating r and θ and t, we have the position of the satellite in the plane of its orbit. However in order to know the position of the satellite in space it is necessary to know the position of the orbital plane in space. This requires the definition of a suitable coordinate system. The coordinate system that is used was first devised by astonomers to describe the positions of the planets and stars. Such a coordinate system is fixed in space and is know as an inertial coordinate system. Figure 4 The right ascension - declination coordinate system 23

24 An inertial coordinate system The system used is the right ascension - declination coordinate system, which is shown in figure 4. In this system the z axis is aligned with the Earth s spin axis. The x axis points from the centre of the Earth to the sun at the vernal equinox (from the Latin vernalis meaning Spring - obviously defined with respect to the Northern Hemisphere). Figure 5 Coordinates used in the right ascension - declination coordinate system The vernal equinox is when the sun s declination is crossing from the southern to the northern hemisphere (normally around 21 March). The y axis makes up a right-handed coordinate system. In this system, the declination of a point in space is its angular displacement measured northward from the equatorial plane, and the right ascension is the angular displacement, measured anticlockwise from the x axis, of the projection of the point in the equatorial plane, these angles are shown in figure 5. 24

25 Classical orbital elements Rotation of Earth r Satellite orbit Sub-satellite track! " # i Perigee of orbit Vernal equinox Ascending node Equator 25

26 Classical orbital elements Rotation of Earth r Satellite orbit Sub-satellite track! " # i Perigee of orbit Vernal equinox Ascending node Equator From figure 6 it can be seen that three angles are used to position an elliptical orbit in the right ascension declination coordinate system. These are the inclination angle, i, the right ascension of the ascending node, Ω, and the argument of perigee, ω. The inclination angle is zero if the orbital plane is in the earth s equatorial plane and the orbit rotates in the same direction of the earth (prograde). If the inclination angle exceeds 90 then the orbit of the satellite is opposite to the rotation of the earth and the orbit is said to be retrograde. NOAA satellites are normally in retrograde orbitswith an inclination angle 99, whereas the Geostationary satellites which have a nominal inclination angle of 0 are in prograde orbits. The ascending node is the point where the satellite crosses the equatorial plane in a northbound direction and the descending node is when the satellite crosses the equatorial plane in the southbound direction. The right ascension of the ascending node (Ω) is measured in the equatorial plane from the x axis to the ascending node, as shown in figure 6. This is also the right ascension of the intersection of the orbital plane of the satellite and the equatorial plane of the earth. The argument of perigee, ω, is the angle measured in the orbital plane between the ascending node and the perigee. The parameters are know collectively as classical orbital elements and are summarised in Table 1. They are measured to obtain the satellites exact position. The parameters with a subscript o indicates the value of the parameter at the epoch time, as these elements change with time. 26

27 Orbits 27

28 Polar orbits 28

29 Gestationary 29

30 Advantages and disadvantages of different orbits 30

31 Track of retrograde polar-orbiting satellite Figure 8 Track of a low near polar orbiting satellite 31

32 Track of retrograde polar-orbiting satellite 32