Lecture 1a: Satellite Orbits

Size: px

Start display at page:

Download "Lecture 1a: Satellite Orbits"

Louisa Cole
5 years ago
Views:

1 Lectue 1a: Satellite Obits

2 Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Calculating satellite obital paametes (assuming cicula motion)

3 Scala & Vectos Scala: a physical quantity that can be chaacteized by a single value, e.g. tempeatue. Scala algeba (addition, subtaction, multiplication, and division) is staightfowad Z c Vecto: a physical quantity that has both magnitude and diection, e.g. displacement, foce, etc. a i k j b Y = ai + b j + ck X

4 Scala & Vectos (cont d) Vecto addition & subtaction: = 3 paallelogam Scala poduct of two vectos (dot poduct) q = 1 2 cosθ Dot poduct yields a scala 1 2 Vecto poduct of two vectos (aka, coss poduct) 1 2 = 1 2 sinθ Vecto poduct yields a vecto 1 q 2 Diection follows the ight-hand ule.

SI basic and deived Units 1. Mass: kilogam (kg) 2. Length: mete (m) 3. Time: second (s) 4. Plane angle: adian (ad; 360 0 = 2p) 5. Speed: length / time (m/s) 6. Acceleation: speed / time (m/s 2 ) 7.

5 SI basic and deived Units 1. Mass: kilogam (kg) 2. Length: mete (m) 3. Time: second (s) 4. Plane angle: adian (ad; = 2p) 5. Speed: length / time (m/s) 6. Acceleation: speed / time (m/s 2 ) 7. Foce: mass x acceleation (kg m/s 2 o N) 8. Enegy: foce x distance (N m o Joule) 9. Powe: enegy / time (N m s -1 o Watt) 10. Pessue: foce / aea (N m -2 o Pascal) 11. Fequency: 1/time (s -1 ) 12. Tempeatue: Kelvin (K; 0 0 C= K)

6 Coodinate Systems Catesian (x,y,z) Pola (, q) zenith Spheical (, q, f) azimuth

7 Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Calculating satellite obital paametes (assuming cicula motion)

8 Newton s Laws of Motion Fist Law: Evey object tends to emain at est o in unifom motion in a staight line, unless acted upon by extenal influences. Isaac Newton Also known as, law of inetia.

9 Newton s Laws of Motion Second Law: The ate of change of momentum (mv) is equal to the impessed foce and takes place in the line in which the foce acts. Isaac Newton F = d(mv) dt = m dv dt = ma F = m dv dt = m d dt ( d dt ) = m d 2 dt 2 whee F is foce, m is mass, v is velocity, a is acceleation, is displacement, and t is time. The aow hat means vecto.

10 Newton s Laws of Motion Thid Law: Action and eaction ae equal and opposite. Isaac Newton Because acceleation is invesely popotional to mass, the bigge ball is affected less by this collision than does the smalle ball. F m = dv dt = a

Newton s Law of Univesal Gavitation Thee is a foce of attaction between any two point masses m1 and m2, which is diectly popotional to the poduct of the masses and invesely popotional

11 Newton s Law of Univesal Gavitation Thee is a foce of attaction between any two point masses m1 and m2, which is diectly popotional to the poduct of the masses and invesely popotional to the squae of thei distance apat 12. m1m2 F =G 2 12 Isaac Newton Whee G=6.67 x N m2 kg-2 m2 12 Reputed descendant of Newton s apple tee in Cambidge m1 Eath Satellite

12 Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Calculating satellite obital paametes (assuming cicula motion) F = ma F = G m 1 m

13 Acceleation in cicula motion v Centipetal acceleation: a = v 2 M Conside cicula motion fo simplicity a = dv dt m q dv v vdt v v dv v = vdt = θ dv = v 2 dt a = dv dt = v 2

14 Centipetal acceleation: a = v 2 (1) M v m Univesal gavitation: F = G Mm!" " F = ma 2 (2) So, the close a satellite is obiting the Eath, the faste it moves (G and M ae constants). Fom (1) & (2): v = GM G = 6.67 x N m 2 kg -2 M = 5.97 x kg e = x 10 6 m (adius of the Eath)

15 v = GM M v m Centipetal acceleation: a = v 2 Univesal gavitation: F = G Mm 2 (1) (2) Peiod of a cicula motion: T = 2π v (3) Fom (1), (2) & (3): T 2 = 4π 2 3 GM

16 Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Keple s Laws 4. Putting Newton s and Keple s Laws togethe and applying them to the Eath-satellite system 5. Oientation of obit in space

17 Keple s Laws Keple s Laws summaize the obsevational facts concening the movements of the heavenly bodies; they wee collected befoe Newton s time (Newton was bon in 1642, 12 ys afte Keple died). Newton s Laws explain why planets behave the way they do. Johannes Keple Fist Law: All planets tavel in elliptical paths with the Sun at one focus. (Note: the laws concening the Sun-planet system applies to the Eath-satellite system as well.)

18 Ellipse Geomety 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 Catesian coodinates x 2 a 2 + y 2 b 2 =1 a: semimajo; b: semimino; e: eccenticity (0-1) e = 1 b2 a 2 Cicle is a special case of ellipse with a=b, so e=0 and the equation becomes x 2 + y 2 = a 2

19 Keple s Laws (cont d) Second Law: The adius vecto fom the Sun to a planet sweeps out equal aeas in equal time. Johannes Keple Sun Planet Planet Thid Law: The atio of the squae of the peiod of evolution of a planet to the cube of its semimajo axis is the same fo all planets evolving aound the Sun. (Again, think of the Eath-satellite system)

Lecture 1a: Satellite Orbits

Lecture 1a: Satellite Orbits Lectue 1a: Satellite Obits Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Univesal Gavita3on 3. Calcula3ng satellite obital paametes (assuming cicula mo3on) Scala & Vectos Scala: a physical quan3ty