Lecture 1a: Satellite Orbits

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Albert Palmer
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1 Lecture 1a: Satellite Orbits

3 Meteorological Satellite Orbits LEO view GEO view Two main orbits of Met Satellites: 1) Geostationary Orbit (GEO) 1) Low Earth Orbit (LEO) or polar orbits

4 Orbits of meteorological satellites Satellites are not free to travel any path but must follow certain orbits dictated by some fundamental laws of physics. Knowing these laws is critical to the understanding (and design) of satellite orbits.

5 Outline 1. Newton s Laws of Motion 2. Newton s Law of Universal Gravitation 3. Calculating satellite orbital parameters (assuming circular motion)

6 Scalar & Vectors Scalar: a physical quantity that can be characterized by a single value, e.g. temperature. Scalar algebra (addition, subtraction, multiplication, and division) is straightforward Z r c Vector: a physical quantity that has both magnitude and direction, e.g. displacement, force, etc. a i k j b Y r = ai + b j + ck X

Scalar & Vectors (cont d) Vector addition & subtraction: r1 r2 r3 r1 + r2 = r3 parallelogram Scalar product of two vectors (dot product) q r2 r1 r2 = r1 r2 cosθ Dot

7 Scalar & Vectors (cont d) Vector addition & subtraction: r1 r2 r3 r1 + r2 = r3 parallelogram Scalar product of two vectors (dot product) q r2 r1 r2 = r1 r2 cosθ Dot product yields a scalar r1 r2 Vector product of two vectors (aka, cross product) r1 r2 = r1 r2 sinθ Direction follows the righthand rule. r1 q r2 Vector product yields a vector

SI basic and derived Units 1. Mass: kilogram (kg) 2. Length: meter (m) 3. Time: second (s) 4. Plane angle: radian (rad; 360 0 = 2p) 5. Speed: length / time (m/s) 6.

8 SI basic and derived Units 1. Mass: kilogram (kg) 2. Length: meter (m) 3. Time: second (s) 4. Plane angle: radian (rad; = 2p) 5. Speed: length / time (m/s) 6. Acceleration: speed / time (m/s 2 ) 7. Force: mass x acceleration (kg m/s 2 or N) 8. Energy: force x distance (N m or Joule) 9. Power: energy / time (N m s -1 or Watt) 10. Pressure: force / area (N m -2 or Pascal) 11. Frequency: 1/time (s -1 ) 12. Temperature: Kelvin (K; 0 0 C= K)

9 Coordinate Systems Cartesian (x,y,z) Polar (r, q) zenith Spherical (r, q, f) azimuth

10 Outline 1. Newton s Laws of Motion 2. Newton s Law of Universal Gravitation 3. Calculating satellite orbital parameters (assuming circular motion)

11 Newton s Laws of Motion First Law: Every object tends to remain at rest or in uniform motion in a straight line, unless acted upon by external influences. Isaac Newton Also known as, law of inertia. Not a no-brainer at all. It s easy to understand that without external force, an object will stay at rest, but it s not obvious (based on daily experience) that without force, an object will keep moving. In fact, ancient Greek philosophers (e.g., Aristotle) gave a different (and wrong) view on this.

12 Newton s Laws of Motion Second Law: The rate of change of momentum (mv) is equal to the impressed force and takes place in the line in which the force acts. Isaac Newton F = d(mv) dt = m dv dt = ma F = m dv dt = m d dt ( dr dt ) = m d 2 r dt 2 where F is force, m is mass, v is velocity, a is acceleration, r is displacement, and t is time. The arrow hat means vector. This law is very powerful because it states that once forces (F) are known, we can predict the motion (r) of an object (just integrate the equation). Inversely, once we know the motion of an object (r), we can infer the net force (F) acted upon it (just differentiate the equation).

13 Newton s Laws of Motion Third Law: Action and reaction are equal and opposite. Isaac Newton Because acceleration is inversely proportional to mass, the bigger ball is affected less by this collision than does the smaller ball. F m = dv dt = a

Newton s Law of Universal Gravitation There is a force of attraction between any two point masses m 1 and m 2, which is directly proportional to the product of the masses and inversely proportional

14 Newton s Law of Universal Gravitation There is a force of attraction between any two point masses m 1 and m 2, which is directly proportional to the product of the masses and inversely proportional to the square of their distance apart r 12. Isaac Newton F = G m 1m 2 r 12 2 Where G=6.67 x N m 2 kg -2 m 2 Satellite r 12 Reputed descendant of Newton s apple tree in Cambridge m 1 Earth

15 Outline 1. Newton s Laws of Motion 2. Newton s Law of Universal Gravitation 3. Calculating satellite orbital parameters (assuming circular motion) F = ma F = G m 1 m 2 r 12 2

16 Acceleration in circular motion v Centripetal acceleration: a = v 2 r M Consider circular motion for simplicity a = dv dt r m q dv v r vdt v v dv v = vdt r = θ dv = v 2 dt r a = dv dt = v 2 r

17 Centripetal acceleration: a = v 2 r (1) M r v m Universal gravitation: F = G Mm!" " F = ma r 2 (2) So, the closer a satellite is orbiting the Earth, the faster it moves (G and M are constants). From (1) & (2): v = GM r If a satellite is flying right near the surface, its speed is G = 6.67 x N m 2 kg -2 M = 5.97 x kg r e = x 10 6 m (radius of the Earth) 0.9 km/s 7.9 km/s 100 km/s What is the speed for LEOs flying at ~ 800 km altitude? 7.5 km/s

0 latitude: G = 6.67 x 10-11 N m 2 kg -2 M = 5.97 x 10 24 kg r e = 6.

18 Centripetal acceleration: a = v 2 Universal gravitation: r F = G Mm r 2 (1) (2) v = GM r Calculate the rotation speed at the equator and at 45 0 latitude: G = 6.67 x N m 2 kg -2 M = 5.97 x kg r e = x 10 6 m (radius of the Earth) 2πr T = = 463 m /s

19 From (1) & (2): v = GM r M r v m Centripetal acceleration: a = v 2 Universal gravitation: r F = G Mm r 2 (1) (2) Period of a circular motion: T = 2πr v (3) From (1), (2) & (3): T 2 r = 4π 2 3 GM This is a powerful conclusion. The period and altitude of satellites are related to each other: higher orbit means longer period

20 T 2 r = 4π 2 3 GM M r v m Calculate the following: 1) For a satellite that has the same period as the Earth s rotation (geostationary satellite), what s the altitude of the orbit (convert m to km)? ~36,000 km 2) For the operational polar orbiting satellites whose altitude is 856 km, calculate the period (convert sec to min or hr). ~ 110 min G = 6.67 x N m 2 kg -2 M = 5.97 x kg r e = x 10 6 m (radius of the Earth)

21 Outline 1. Newton s Laws of Motion 2. Newton s Law of Universal Gravitation 3. Kepler s Laws 4. Putting Newton s and Kepler s Laws together and applying them to the Earth-satellite system 5. Orientation of orbit in space

22 Kepler s Laws Kepler s Laws summarize the observational facts concerning the movements of the heavenly bodies; they were collected before Newton s time (Newton was born in 1642, 12 yrs after Kepler died). Newton s Laws explain why planets behave the way they do. Johannes Kepler First Law: All planets travel in elliptical paths with the Sun at one focus. (Note: the laws concerning the Sun-planet system applies to the Earth-satellite system as well.)

23 Ellipse Geometry Ellipse is a locus of points in a plane such that the sum of the distances to two fixed points (foci) is a constant. In Cartesian coordinates x 2 a 2 + y 2 b 2 =1 a: semimajor; b: semiminor; e: eccentricity (0-1) e = 1 b2 a 2 Circle is a special case of ellipse with a=b, so e=0 and the equation becomes x 2 + y 2 = a 2

24 Kepler s Laws (cont d) Second Law: The radius vector from the Sun to a planet sweeps out equal areas in equal time. Johannes Kepler Sun Planet Planet Third Law: 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. (Again, think of the Earth-satellite system)

25 Kepler s Laws Newton s Laws F = m dv dt = ma F = G m 1m 2 r 12 2 T 2 r = 4π 2 3 GM First Law: All planets travel in elliptical paths with the Sun at one focus. Second Law: The radius vector from the Sun to a planet sweeps out equal areas in equal time. Third Law: 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.

Lecture 1a: Satellite Orbits

Lecture 1a: Satellite Orbits Lectue 1a: Satellite Obits Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Calculating satellite obital paametes (assuming cicula motion) Scala & Vectos Scala: a physical quantity