Lecture 1a: Satellite Orbits
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1 Lectue 1a: Satellite Obits
2
3 Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Univesal Gavita3on 3. Calcula3ng satellite obital paametes (assuming cicula mo3on)
4 Scala & Vectos Scala: a physical quan3ty that can be chaacteized by a single value, e.g. tempeatue. Scala algeba (addi3on, subtac3on, mul3plica3on, and division) is staighnowad Z c Vecto: a physical quan3ty that has both magnitude and diec3on, e.g. displacement, foce, etc. a i k j b Y = ai + b j + ck X
5 Scala & Vectos (cont d) Vecto addi3on & subtac3on: = 3 paallelogam Scala poduct of two vectos (dot poduct) θ = 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 θ 2 Diec3on follows the ight-hand ule.
6 SI basic and deived Units 1. Mass: kilogam (kg) 2. Length: mete (m) 3. Time: second (s) 4. Plane angle: adian (ad; = 2π) 5. Speed: length / 3me (m/s) 6. Accelea3on: speed / 3me (m/s 2 ) 7. Foce: mass x accelea3on (kg m/s 2 o N) 8. Enegy: foce x distance (N m o Joule) 9. Powe: enegy / 3me (N m s -1 o Wae) 10. Pessue: foce / aea (N m -2 o Pascal) 11. Fequency: 1/3me (s -1 ) 12. Tempeatue: Kelvin (K; 0 0 C= K)
7 Coodinate Systems Catesian (x,y,z) Pola (, θ) zenith Spheical (, θ, φ) azimuth
8 Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Univesal Gavita3on 3. Calcula3ng satellite obital paametes (assuming cicula mo3on)
9 Newton s Laws of Mo3on Fist Law: Evey object tends to emain at est o in unifom mo3on in a staight line, unless acted upon by extenal influences. Isaac Newton Also known as, law of ine3a. Not a no-baine at all. It s easy to undestand that without extenal foce, an object will stay at est, but it s not obvious (based on daily expeience) that without foce, an object will keep moving. In fact, ancient Geek philosophes (e.g., Aistotle) gave a diffeent (and wong) view on this.
10 Newton s Laws of Mo3on 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 accelea3on, is displacement, and t is 3me. The aow hat means vecto. This law is vey poweful because it states that once foces (F) ae known, we can pedict the mo3on () of an object (just integate the equa3on). Invesely, once we know the mo3on of an object (), we can infe the net foce (F) acted upon it (just diffeen3ate the equa3on).
11 Newton s Laws of Mo3on Thid Law: Ac3on and eac3on ae equal and opposite. Isaac Newton Because accelea3on is invesely popo3onal to mass, the bigge ball is affected less by this collision than does the smalle ball. F m = dv dt = a
12 Newton s Law of Univesal Gavita3on Thee is a foce of aeac3on between any two point masses m 1 and m 2, which is diectly popo3onal to the poduct of the masses and invesely popo3onal to the squae of thei distance apat 12. Isaac Newton F = G m 1 m Whee G=6.67 x N m 2 kg -2 m 2 Satellite 12 Reputed descendant of Newton s apple tee in Cambidge m 1 Eath
13 Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Univesal Gavita3on 3. Calcula3ng satellite obital paametes (assuming cicula mo3on) F = ma F = G m 1 m
14 Accelea2on in cicula mo2on v Centipetal accelea3on: a = v 2 M Conside cicula mo3on fo simplicity a = dv dt m θ dv v vdt v v dv v = vdt = θ dv = v 2 dt a = dv dt = v 2
15 Centipetal accelea3on: a = v 2 (1) M v m Univesal gavita3on: F = G Mm!" " F = ma 2 (2) So, the close a satellite is obi3ng the Eath, the faste it moves (G and M ae constants). Fom (1) & (2): v = GM If a satellite is flying ight nea the suface, its speed is G = 6.67 x N m 2 kg -2 M = 5.97 x kg e = x 10 6 m (adius of the Eath) 0.9 km/s 7.9 km/s 100 km/s What is the speed fo LEOs flying at ~ 800 km al3tude? 7.5 km/s
16 Centipetal accelea3on: a = v 2 Univesal gavita3on: F = G Mm 2 (1) (2) v = GM Calculate the ota3on speed at the equato and at 45 0 la3tude: G = 6.67 x N m 2 kg -2 M = 5.97 x kg e = x 10 6 m (adius of the Eath) 2π T = = 463 m /s
17 v = GM M v m Centipetal accelea3on: a = v 2 Univesal gavita3on: F = G Mm 2 (1) (2) Peiod of a cicula mo3on: T = 2π v (3) Fom (1), (2) & (3): T 2 = 4π 2 3 GM This is a poweful conclusion. The peiod and al3tude of satellites ae elated to each othe: highe obit means longe peiod
18 T 2 = 4π 2 3 GM M v m Calculate the following: 1) Fo a satellite that has the same peiod as the Eath s otation (geostationay satellite), what s the altitude of the obit (convet m to km)? ~36,000 km 2) Fo the opeational pola obiting satellites whose altitude is 856 km, calculate the peiod (convet sec to min o h). ~ 110 min G = 6.67 x N m 2 kg -2 M = 5.97 x kg e = x 10 6 m (adius of the Eath)
19 Outline 1. Newton s Laws of Mo3on 2. Newton s Law of Univesal Gavita3on 3. Keple s Laws 4. Puong Newton s and Keple s Laws togethe and applying them to the Eath-satellite system 5. Oienta3on of obit in space
20 Keple s Laws Keple s Laws summaize the obseva8onal facts concening the movements of the heavenly bodies; they wee collected befoe Newton s 3me (Newton was bon in 1642, 12 ys ape Keple died). Newton s Laws explain why planets behave the way they do. Johannes Keple Fist Law: All planets tavel in ellip3cal paths with the Sun at one focus. (Note: the laws concening the Sun-planet system applies to the Eath-satellite system as well.)
21 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 equa3on becomes x 2 + y 2 = a 2
22 Keple s Laws (cont d) Second Law: The adius vecto fom the Sun to a planet sweeps out equal aeas in equal 3me. Johannes Keple Sun Planet Planet Thid Law: The a3o of the squae of the peiod of evolu3on 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)
23 Keple s Laws Newton s Laws F = m dv dt = ma F = G m 1 m T 2 = 4π 2 3 GM Fist Law: All planets tavel in ellip3cal paths with the Sun at one focus. Second Law: The adius vecto fom the Sun to a planet sweeps out equal aeas in equal 3me. Thid Law: The a3o of the squae of the peiod of evolu3on of a planet to the cube of its semimajo axis is the same fo all planets evolving aound the Sun.
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