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 in position of n object reltive to more distnt bckground when the object is viewed from different ngles. The Erth s orbitl motion round the Sun results in us observing nerby strs from slightly different ngles t different times of yer. Therefore, these strs exhibit nnul prllx when viewed ginst the bckground of more distnt objects. Imge credit: Liverpool John Moores University Mrch d X September NOT TO SCLE distnt bckground strs The prllx ngle of str X is given by tn = We cn mke use of the smll ngle pproximtions: if á 1 (nd is expressed in RDINS), then d 3 ( 3! + K) 2 ( 1 2! + K) 1 sin [3.1] cos [3.2] tn = sin cos 1 [3.3] 1 Positionl stronomy Pge 7 Lecture 2
So for the prllx ngle, or d = tn d = [3.4] For the Erth s orbit, = 1 U. Suppose = 1 rc second. 1 degree = 60 minutes of rc 1 minute of rc = 60 seconds of rc 1 = 60 =3600 Then (nd expressing = 1 in rdins) d = = 2π 1U ( 360 60 60) 206265U This distnce d is clled prllx second or prsec (pc). 1 pc = 3.086 10 16 m = 3.26 light yers Definition: 1 prsec is the distnce t which length of 1 U subtends n ngle of 1 rc second. So if is mesured in rc seconds, nd Erth s orbitl rdius is mesured in U, then the prllx distnce d is given in prsecs by eqution (3.4). Usully, prllx mesurements re used this wy round to determine stellr distnces. For Proxim Centuri t 4.24 ly (= 1.3 pc), = 1 / 1.3 = 0.77 Prllx ngle decreses for strs t incresing distnce, so no strs will show prllx shift of greter thn bout 1, which is only 1/1296000 of the full circumference of the sky. The vst mjority of strs re so fr wy tht their prllx shift is completely imperceptible with current instruments. 1 Positionl stronomy Pge 8 Lecture 2
4. The Celestil Sphere Of course, strs re not ctully fixed in position they move reltive to the Sun. This movement cn be seprted into component long our line of sight (i.e. motion towrds or wy from us), which is clled rdil velocity, nd motion perpendiculr to our line of sight (i.e. in the plne of the sky), which is clled proper motion. However, becuse the strs re t such gret distnces, even if their speed reltive to the Sun is quite lrge (~ 100 km/s for nerby strs), their proper motion, seen s shift in ngulr position on the sky, is still very smll. The str with the lrgest proper motion is rnrd s Str, which chnges ngulr position by 10.3 per yer. So the pprent ngulr position of the strs chnges by less thn 1 prt in 100,000 due to proper motion, nd by less thn 1 prt in 1,000,000 due to the prllx from Erth s orbitl motion: to very good pproximtion, we cn suppose tht the strs re ll fixed in position on the surfce of n imginry sphere, centred on the observer, nd with very lrge rdius. Thus, the only informtion we need to mp str s pprent loction is its ngulr position its distnce wy nd the Erth s position in its orbit re unimportnt. We cll this imginry surfce of strs the celestil sphere. The centre of the celestil sphere is lwys the observer. 1 Positionl stronomy Pge 9 Lecture 2
5. Elements of the Sphere Definition: sphere is the surfce of ll points which re equidistnt from one point (the centre, O). Definition: plne which psses through the centre of the sphere cuts the sphere in gret circle. Definition: plne which does not pss through the centre of the sphere cuts the sphere in smll circle. Definition: The dimeter pssing through the centre of the sphere perpendiculr to circle cuts the sphere t two dimetriclly opposite points. These points re the poles (N, S) of the circle. N smll circle dimeter C O D gret circle S Theorem: Two points on the sphere which re not dimetriclly opposite (, ) re joined by unique gret circle. The shortest distnce on the surfce of the sphere between these points is the (shorter) gret circle rc. (Gret circle rcs ply the role of stright lines on spheres.) Theorem: If two points on the sphere re dimetriclly opposite (C, D), every gret circle which psses through one point lso psses through the other. 1 Positionl stronomy Pge 10 Lecture 2
The length l of n rc subtended by n ngle in circle of rdius r is given by where is given in rdins. l = r [5.1] The rdius r of the celestil sphere is presumbly extremely lrge by terrestril stndrds. However, we cn choose to express it in whtever units we like (metres, miles, light yers, gigprsecs ). Without concerning ourselves with wht the vlue of the rdius might ctully be (since we showed in Section 4 tht it doesn t mtter), we cn choose the units so tht r is exctly one. Therefore, by eqution (5.1), gret circle rc lengths on the celestil sphere tke the sme vlue s the ngle they subtend t the centre of the sphere. We cn therefore use gret circle rc lengths nd the ngle they subtend t the sphere centre interchngebly. N l ψ r E x ω F O r ψ O Since the rdius of the sphere is unity, the smll circle with centre O will hve rdius of S r = 1 sinψ = sinψ so the smll circle rc length EF will be given by x = ω sinψ [5.2] 1 Positionl stronomy Pge 11 Lecture 2
N J G O H 90 K Theorem: The sphericl ngle between two gret circles is the ngle between their plnes t the centre of the sphere O, or equivlently the ngle between their tngents t their points of intersection (G, H). HK = KG = HJ = JG = 90 nd JK = Definition: Three points, nd C on the surfce of sphere (nd not ll lying on single gret circle) define unique smll circle., nd C re the vertices of sphericl tringle, the sides of which re gret circle rcs. S c b C C y convention, we will lbel the sphericl tringle vertex ngles s, nd C nd the opposite sides s, b, nd c. Note tht the internl ngles of sphericl tringle do NOT dd up to 180. 1 Positionl stronomy Pge 12 Lecture 2