Lecture 2c: Satellite Orbits

Transcription

1 Lecture 2c: Satellite Orbits

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4 Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Universal Gravita3on 3. Kepler s Laws 4. Pu>ng Newton and Kepler s Laws together and applying them to the Earth-satellite system 5. Orienta3on of orbit in space 6. Orbital parameters & orbit perturba3on 7. Satellite posi3oning, tracking & naviga3on 8. Orbits of Meteorological Satellites (MetSat)

5 Orbit Perturba3on Forces Minor Effect Force Nonspherical gravitational field Gravitational attraction of other bodies Radiation pressure Particle flux Lift and drag Electromagnetic forces Source Nonspherical, nonhomogeneous Earth Sun, moon, planets Solar radiation Solar wind Residual atmosphere Interaction of electrical currents in the satellite with Earth s magnetic field

6 One can think of the Earth as a sphere with a 21-km-thick waist belt around the equator. In addi3on to the gravita3onal force that points to the center of the Earth, there is an addi3onal force that drags the satellite toward the equator. Equator 6357 km Pole 6378 km Think-Pair-Share: what kind of effect do you think this extra belt will have on an orbiting satellite?

7 where p = mv Torque (τ): a vector that measures the tendency of a force to rotate an object about some axis. τ r F Angular momentum (L): a vector that measures rota3on. r mv F r r F τ = r F = r d(mv) dt ana log y :τ! = dl"! dt = d(r mv) dt F!" = d(mv " ) dt = dl dt (right-hand rule): torques points from the slide toward us. Consequently, the top will not fall but wobbles (precession).

8 Similarly, this equatorial belt does not make satellite to fall to the equatorial plane, but changes the right ascension of the ascending node (Ω) and the argument of perigee (ω). a, i and ε are virtually unperturbed.

9 Ω: when i < 90 0, dω/dt < 0; when i > 90 0, dω/dt > 0 ω: it does not concern us much because most orbits are circular dω dt = n 3 2 J 2 r ee a 2 ( 1 ε 2 ) 2 cosi M: when 2 5 >0, i.e., 2 sin2 i i < 57 0, satellite orbits slightly faster than it would in an unperturbed orbit. dω dt = n 3 2 J 2 r ee a 2 ( 1 ε 2 ) sin2 i dm dt = n = n J 2 r ee a 2 ( 1 ε 2 ) 3 / sin2 i Where r ee is equatorial radius of the Earth, J 2 is a constant

10 We can take advantage of the wobbling Ω: when i < 90 0, dω/dt < 0; when i > 90 0, dω/dt > 0 dω dt = n 3 2 J 2 r ee a 2 ( 1 ε 2 ) 2 cosi

11 Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Universal Gravita3on 3. Kepler s Laws 4. Pu>ng Newton and Kepler s Laws together and applying them to the Earth-satellite system 5. Orienta3on of orbit in space 6. Orbital parameters & orbit perturba3on 7. Satellite posi3oning, tracking & naviga3on 8. Orbits of Meteorological Satellites (MetSat)

12 Satellite Posi3oning Satellite posi3oning is to locate a satellite in a perturbed orbit at 3me t, given the orbital elements. Element Semimajor axis* Eccentricity Inclination Argument of perigee Right ascension of ascending node Mean anomaly** Epoch time Symbol a ε i ω o Ω o M o t o You only need 3 parameters (x/y/z or r/δ/ω to posi3on a point in 3D space

13 Procedure 1. Find the orbital elements of the satellite. 2. Update the variable elements (M, Ω, ω) to the time (t ) that you are interested in: they are affected by non spherical gravitational field of the Earth.

14 Procedure 1. Find the orbital elements of the satellite. 2. Update the variable elements (M, Ω, ω) to the time (t ) that you are interested in: they are affected by non spherical gravitational field of the Earth. 3. Use Kepler s equation to convert from mean anomaly (M) to true anomaly (θ). r = a(1 ε2 ) 4. Use ellipse equation 1+ εcosθ to calculate r, the distance of the satellite from the center of the earth.

15 Procedure 1. Find the orbital elements of the satellite. 2. Update the variable elements (M, Ω, ω) to the time (t ) that you are interested in: they are affected by non spherical gravitational field of the Earth. 3. Use Kepler s equation to convert from mean anomaly (M) to true anomaly (θ). r = a(1 ε2 ) 4. Use ellipse equation 1+ εcosθ to calculate r, the distance of the satellite from the center of the earth.

16 Procedure 1. Find the orbital elements of the satellite. 2. Update the variable elements (M, Ω, ω) to the time (t ) that you are interested in: they are affected by non spherical gravitational field of the Earth. 3. Use Kepler s equation to convert from mean anomaly (M) to true anomaly (θ). r = a(1 ε2 ) 4. Use ellipse equation 1+ εcosθ to calculate r, the distance of the satellite from the center of the earth. 5. Once r and θ are known and (i, Ω, ω) are determined, the procedure is straightforward (although not trivial) to position the satellite in terms of (r,ω,δ) or (x,y,z).

17 Procedure 1. Find the orbital elements of the satellite. 2. Update the variable elements (M, Ω, ω) to the time (t ) that you are interested in: they are affected by non spherical gravitational field of the Earth. 3. Use Kepler s equation to convert from mean anomaly (M) to true anomaly (θ). r = a(1 ε2 ) 4. Use ellipse equation 1+ εcosθ to calculate r, the distance of the satellite from the center of the earth. 5. Once r and θ are known and (i, Ω, ω) are determined, the procedure is straightforward (although not trivial) to position the satellite in terms of (r,ω,δ) or (x,y,z). This will be your in-class lab 1 (Feb 22) and part of homework 1 (read )

18 Read Section This involves rotation of vectors First, place the ellipse on x-y plane (input: semimajor, eccentricity, and mean anomoly) Second, do a series of vector rota3ons (input: ω, Ω, i) Rotate with respect to z axis Rotate with respect to z axis Rotate with respect to x axis

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21 If you prefer latitude/longitude, then La3tude = δ Longitude = Ω SAT - Ω Greenwich

22 La3tude = δ Longitude = Ω SAT - Ω Greenwich This term rotates at the same angular speed as the Earth: 7.29 x 10-5 radian/sec

23 Satellite Tracking To track a satellite, one needs to be able to point one s antenna at it. This requires calculating the local azimuth and elevation angles for the satellite (note this is not the right-ascension declination coordinate, but a local coordinate).

24 ζ! r s = x s y s z s r s cosθ s cosψ s = r s cosθ s sinψ s r s sinθ s ψ s θ s! r e = x e y e z e r e cosθ e cosψ e = r e cosθ e sinψ e r e sinθ e θ s and ψ s are determined from satellite positioning. θ e and ψ e are for the local position (latitude and longitude). cosζ =! r D = r! s r! e! r e r! D r! e r! D This is the direction of the antenna to point to the satellite.

25 Satellite Naviga8on In addition to knowing where a satellite is in its orbit, we also need to know the Earth coordinate (lat/lon) of the particular scene the satellite is viewing navigation. We need the following information: 1) satellite position, 2) attitude of the satellite (its orientation), and 3) the scanning geometry.