20. Direct and Retrograde Motion

Size: px

Start display at page:

Download "20. Direct and Retrograde Motion"

Kathryn Cook
6 years ago
Views:

1 0. Direct nd Retrogrde Motion When the ecliptic longitude λ of n object increses with time, its pprent motion is sid to be direct. When λ decreses with time, its pprent motion is sid to be retrogrde. ince the simplified solr motion hs the un move uniformly round the ecliptic once per yer, the motion of the un is lwys direct. The plnets (nd other solr system objects) hve pprent motion which is sometimes direct nd t other times retrogrde. Inferior plnets: Consider plnet closer to the un thn is the rth. When the plnet is on the opposite side of the un to the rth (i.e. round its superior conjunction), both the orbitl motion of the plnet nd of the rth ct in the sme sense reltive to the distnt bckground strs. Therefore, the motion of the plnet is direct. However, when the plnet is on the sme side of the un s the rth (i.e. round its inferior conjunction), since the plnet s orbitl velocity is greter thn the rth s it overtkes the rth on the inside. The plnet s orbitl velocity cncels out (nd more) tht of the rth, mking its pprent motion reltive to the distnt strs be retrogrde in chrcter. A ositionl Astronomy ge 6 Lecture

2 uperior plnets: Consider now plnet further wy from the un thn is the rth. For most of its synodic period, this plnet s pprent motion will be direct. However, ner to its time of opposition, the fster-orbiting rth will overtke it on the inside, leding to period of retrogrde motion for the superior plnet This is seen s retrogrde loop in the plnet s pprent motion in the sky: the plnet s motion is initilly direct, then it slows to hlt t sttionry point, the plnet s direction of motion reverses, it slows to nother hlt t its second sttionry point, nd then it proceeds onwrds in direct motion gin. At the sttionry points, dλ / dt = 0 The pprent motion of Mrs direct sttionry point retrogrde sttionry point Imge credit: Tunc Tezel direct A ositionl Astronomy ge 6 Lecture

3 . longtion of the ttionry oints We hve seen tht when superior plnet is t opposition, with n elongtion of = 80, its motion is retrogrde: the plnet s orbitl velocity vector lies prllel to the rth s orbitl velocity vector, nd since the rth s orbitl velocity v is greter thn the superior plnet s orbitl velocity v, the rth overtkes the superior plnet t opposition, giving it n pprent retrogrde motion reltive to the bckground strs. superior plnet in opposition superior plnet in qudrture v v v v When superior plnet is t qudrture, with n elongtion of = 90, the rth s velocity vector points stright t the plnet. Thus, the rth s orbitl motion mkes no contribution to the pprent movement of the superior plnet ginst the bckground strs, s seen from rth. ince we see only the plnet s own orbitl motion, t qudrture the superior plnet s motion is direct. At some elongtion between 90 nd 80, the plnet must switch over from hving direct motion to hving retrogrde motion. This occurs t the sttionry point, which therefore itself must hve n elongtion lying between 90 nd 80. To find the elongtion of the sttionry point, we must determine where the orbitl velocity of the plnet nd the orbitl velocity of the rth combine to give zero pprent movement of the plnet reltive to the bckground strs. A ositionl Astronomy ge 6 Lecture

4 uppose superior plnet is t one of its sttionry points. uppose lso tht the rth is perfectly motionless. We observe the plnet long the rth-plnet line. We do not discern ny motion long this line of sight, but motion perpendiculr to this line of sight will be seen s n pprent shift of the position of the plnet reltive to the bckground strs. superior plnet t sttionry point v v cos v Thus, the plnet s pprent motion due to its own orbitl velocity is given by the trnsverse component of this velocity, v cos. uppose insted tht the superior plnet is perfectly motionless, so tht ll pprent motion of the plnet is due only to the rth s orbitl motion. The plnet s pprent motion due to rth s orbitl velocity is given by the trnsverse component of this velocity, v cos (80 - ) = - v cos (see ection 0). superior plnet t sttionry point v 80 - v v cos (80 - ) v - 90 A ositionl Astronomy ge 6 Lecture

5 The trnsverse components of the plnet s orbitl velocity nd the rth s orbitl velocity ct in opposing senses. Therefore, they will exctly cncel ech other out (giving no pprent motion of the superior plnet sttionry point) when their mgnitudes re equl: v cos = v o ( ) = v cos cos 80 The orbitl velocity is given by v = ω r where ω is the plnet s orbitl ngulr velocity nd r is its orbitl rdius. Therefore (by eqution 7. nd subsequent) v = ω π = T nd v π = ω = = π T where is the superior plnet s orbitl rdius in AU nd T is its orbitl period in yers. Therefore π cos = π cos T ince we re working in AU nd yers, by Kepler s rd Lw (see Dynmicl Astronomy), = T, so π cos = π cos cos = cos cos = cos cos = cos sin = cos where we use the identity cos x + sin x =. We wish to obtin by eliminting. From eqution (7.), sin = sin A ositionl Astronomy ge 65 Lecture

6 Therefore sin sin = = cos Dividing ll terms by cos we hve tn sec = where sec x = / cos x nd tn x = sin x / cos x. Now, using the identity sec x = + tn x, we isolte tn : tn + tn = tn tn = = tn = ( ) = ( ) = ( )( + ) + We know tht must lie between 90 nd 80, i.e. in the nd qudrnt (see pge 6), so tn is negtive: tn = + Also, since lies in the nd qudrnt, will be 80 minus the principl ngle (clculted using the modulus of tn ), so = 80 o rctn + [.] Thus, the elongtion of the sttionry points of superior plnet depend only on the plnet s orbitl rdius. A ositionl Astronomy ge 66 Lecture

NOT TO SCALE. We can make use of the small angle approximations: if θ á 1 (and is expressed in RADIANS), then

NOT TO SCALE. We can make use of the small angle approximations: if θ á 1 (and is expressed in RADIANS), then 3. Stellr Prllx y terrestril stndrds, the strs re extremely distnt: the nerest, Proxim Centuri, is 4.24 light yers (~ 10 13 km) wy. This mens tht their prllx is extremely smll. Prllx is the pprent shifting