Light Time Delay and Apparent Position

Transcription

1 Light Time Delay and ppaent Position nalytical Gaphics, Inc

2 Contents Intoduction... 3 Computing Light Time Delay... 3 Tansmission fom to... 4 Reception at fom signal sent fom... 4 The Inetial Fame... 4 Choosing an Inetial Fame... 5 Eath Opeations... 6 Signal Path... 6 beation... 6 Stella beation... 7 nnual and Diunal beation... 7 Planetay beation... 8 Optical Measuements of 10

3 Intoduction The elative position of an object with espect to an object can be computed in a vaiety of ways, depending on the model of signal tansmission between the two objects. When light time delay is not consideed, the speed of light is consideed infinite and thee is no time diffeence between the tansmission event and the eception event. Quantities that ignoe light time delay ae often temed tue ; e.g., the tue elative position of with espect to, computed as: () t = R () t R () t (1) The tem appaent is used when the elative position vecto accounts in some way fo light time delay. The appaent position models signal tansmission occuing at the finite speed of light so that a signal tansmitted at time t is not eceived until t+ t, whee t is the light time delay (a positive numbe). Light popagation models have a ich histoy, but fo ou puposes we only need be concened with thee kinematics models: (i) Galilean Relativity, (ii) Special Relativity; and (iii) Geneal Relativity. Galilean Relativity is by fa the most widely known model, whee space is completely sepaable fom the concept of time. Space is modeled as a Euclidean space with the standad vecto opeations fo a linea space; time is an absolute quantity known to all obseves. Special Relativity models light popagation in such a manne that all inetial obseves will measue the speed of light (in vacuum) as the same constant value c. Space is no longe sepaable fom time; space-time is not a Euclidean space but instead a Minkowski space. Concepts that wee once tivial now become moe complicated: diffeent inetial obseves now disagee on simultaneity of events, on distances between objects, and even on how fast time evolves. Howeve, light still popagates as a staight line in the spatial components. Geneal Relativity goes one step futhe, emoving the special status of inetial obseves and intoducing mass as geneating the cuvatue of space-time itself. The light path deflects (cuves) in the spatial components nea massive objects. Ou goal in the modeling of light popagation is simply to include the fist ode coections on Galilean Relativity caused by Special Relativity fo signal tansmission. Thus, we stive fo accuacy to ode β, whee β = v/c, whee v is the inetial velocity of a fame being consideed. The light path then is a staight line in inetial space whee the signal moves at constant speed c (i.e. gavitational deflection is ignoed). Computing Light Time Delay We will fist conside the light time delay fo a signal tansmitted fom an object to an object. Late, we will conside the delay fo a eceived signal at. 3 of 10

4 Tansmission fom to Conside an inetial fame F. Let R locate object in F; let R locate object in F. Let the elative position vecto be defined by () t = R ( t+ t) R () t (2) whee t is the time of tansmission fom and t+ t is the time of eception at. The light time delay is t which will depend on t as well. Let = be the ange between the objects. Then t = /c. One usually knows the locations fo the objects and and computes the light time delay at time t though iteation. Fist, a value of t is guessed (often taken to be 0.0 o the last value computed at a pevious time) and (t) is computed. new value fo t is found fom /c and the pocedue epeats. The iteation stops wheneve the impovement in the estimate to t is less than the light time delay convegence toleance. Typically, few iteations ae equied as the pocedue conveges vey apidly. Reception at fom signal sent fom In this case, the elative position vecto is () t = R ( t t) R () t (3) whee the signal is eceived by at time t. The same pocedue is used to find t, using (t) above. The Inetial Fame NOTE: The light time delay t computed fo the tansmission fom and fo eception at ae diffeent, as is the elative position vecto. The choice of the inetial fame is impotant when computing light time delay, as it will affect the esults. This is a consequence of Special Relativity. Let F and F be two inetial fames with paallel axes. Let v be the velocity of F with espect to F. In Special Relativity, time is not absolute but is instead associated with a fame: let t denote time in F and t denote time in F. Fo simplicity, assume the fames ae coincident at t=0. Then the Loentz tansfomation elating these two coodinate time values is t' γ R = t βi (4) c 4 of 10

5 whee R is the position vecto of a location in F (measued fom the oigin of F), and v 2 1 β=,, 1,and c β = β δ = β γ = (5) δ The value t is the value of time in F fo an event at time t at position R in F. Note that t depends on both t and R. Conside the case of tansmission fom object, located at the oigin of F at time t=0, to object that eceives the signal at time t= t. The value of t at tansmission is computed to be 0 (since both t and R ae zeo then). In F, the light time delay t is computed by solving R ( t) = c t (6) fo t. Using the Loentz tansfomation, the value of t at eception is R R t' = γ t βi = γ t( 1 eˆ ),whee ˆ R i β e = R (7) c c t In F, the light time delay is ( eˆ R β) t' = γ t 1 i (8) To ode β, γ is 1.0, so the diffeence δt in the computed light time delays between the two fames is δ t = t t' = t ( eˆ β) R i (9) The case of eception at at time t=0 is analogous, poducing the same esult. NOTE: The choice of inetial fame affects the computation of both the light time delay t and the appaent elative position vecto. Choosing an Inetial Fame Most space applications involve objects located nea one cental body. It is natual to associate a cental body with an object. The natual inetial fame to use fo modeling spacecaft motion nea a cental body is the inetial fame of the cental body. (We use the tem CI fo Cental ody Inetial.) The CI fame is a natual choice fo the inetial fame fo computing light time delay. Fo objects that ae fa fom thei cental body, howeve, the moe appopiate inetial fame to model motion is a fame with oigin at the sola system baycente. This fame is used to model the motion of the cental bodies themselves. This povides anothe choice fo the inetial fame. 5 of 10

6 We take the view that CI is the pefeable fame fo computing light time delay, but we want to insue that its use appopiately models the physics of the situation at hand. Thus, at the stat of the light time delay computation we compute the diffeence δt between the use of the CI and sola system baycente fames. If this diffeence is less than the light time delay convegence toleance, then eithe fame may be used to obtain the same level of accuacy we choose the CI fame because it is less expensive computationally. If the diffeence is moe than the toleance, we use the sola system baycente fame knowing that it is a bette model of an inetial fame in geneal. Eath Opeations Fo the light time delay convegence toleance of 50.e-5 seconds (i.e., 50 mico-seconds), objects located fom nea the Eath s suface to just outside the geosynchonous belt will use Eath s inetial fame fo pefoming light time delay computations. Fathe out than this, the sola system baycente fame will be used. In paticula, computations involving objects at the Eath-Moon distance will use the sola system baycente fame fo computation of light time delay. Signal Path With t and detemined fom the light time delay computation pefomed in the inetial fame F, it is now possible to model the actual signal tansmission (i.e., the path of the signal though F). The signal path is given by Tansmit fom at t: s( t+ τ ) = R ( t) + cτ e ˆ (10) ( τ) ˆ Receive at at t: s( t+ τ t) = R ( t) + c t e (11) whee 0 τ t andτ=0 locates the tansmission event and τ= t locates the eception event. The appaent diection is given by eˆ =, =, = R( t+ σ t) R ( t) (12) whee t is the light time delay computed in F and σ=1 in the case of tansmission and σ=-1 fo eception. beation beation is the change in the peceived diection of motion caused by the obseve s own motion. The classic example of abeation involves two men out in the ain. One man is stationay and peceives the velocity of the ain as staight down fom ovehead at velocity u. The othe man is walking along the gound at velocity v. In the moving 6 of 10

7 man s fame, the velocity of the ain is u-v. (This is the value as computed using Galilean Relativity; the value accoding to Special Relativity is moe complicated but the conclusions ae the same). This elative velocity makes an angleϕ with the vetical whee 1 v ϕ = tan (13) u The faste the man walks, the lage his peceived deflection of the ain fom the vetical. In technical souces, abeation is usually discussed in the context of eithe stella o planetay abeation. Stella abeation was fist consideed when looking at stas though optical telescopes it is the peceived change in diection of light. Planetay abeation usually efes to two effects combined, light time delay and the peceived change in the diection of light. In both cases, the obseve s velocity elative to the fame in which the light path was computed esults in abeation. Stella beation Typically, stalight is modeled as satuating the sola system with light. The light is consideed to move in a staight line though the sola system. The actual tansmission time at the sta is unmodeled (being moe uncetain than the diection to the sta itself) so light time delay is not consideed. Howeve, abeation caused by an obseve s motion in the sola system as the obseve eceives the light can be computed, and is efeed to as stella abeation. Let the diection to a sta fom an obseve (accounting fo pope motion of the sta and paallax) be e ˆ. Then the appaent diection of the sta, accounting fo stella abeation, is: eˆ + β v pˆ =,wheeβ = (14) eˆ + β c and v is the velocity of the obseve with espect to the sola system baycente fame. The fomula above is the Galilean fomula, equation ( ) 1 ; the Special Relativity fomula is given by ( ) 1 and is simply a use of the Loentz tansfomation fo velocities. The fomula above is accuate to ode β. NOTE: The stella abeation fomula above models the obseve eceiving a signal, not tansmitting a signal. nnual and Diunal beation While the concept of abeation is simple, its computation can be complicated, depending on which factos ae consideed fo detemining the obseve s velocity v with espect to F. stonomes have compatmentalized diffeent aspects of the computation, coining 7 of 10

8 tems fo each aspect s contibution. The tem annual abeation is meant to identify the contibution of the obseve s cental body velocity in the sola system: v= v + v (15) cb whee v cb is the velocity of the cental body with espect to the sola system baycente fame and v /cb is the velocity of the obseve with espect to the cental body. When v cb is used to compute abeation, athe than v, then only the effects of annual abeation have been consideed in the pope appaent elative position. We tem v /cb the diunal abeation (the contibution to v apat fom the cental body motion). In some technical souces, the tem diunal abeation is eseved fo the contibution to v /cb made by the otation of the cental body itself, and othe tems ae used to descibe the othe contibutions to the oveall value of v /cb. /cb Planetay beation Usually, planetay abeation efes to two effects combined: light time delay and the stella abeation (i.e., the change in the peceived diection of motion caused by an obseve s motion). To ode β, the esults can be computed coectly using the simple Galilean fomulas. We have peviously discussed light time delay and detemined a method fo computing the light time delay t, the appaent elative position vecto, and the signal path s by identifying an inetial fame F to pefom the computations. We now conside the effect of abeation. Conside anothe inetial fame F coincident with the obseve at the event time t, whose constant velocity v is the value of the obseve s velocity at time t. ecause the obseve s velocity is not (usually) constant in time, we ll associate a new inetial fame F fo each time t, calling the collection of inetial fames the co-moving inetial fames at. The appaent position of with espect to as peceived by an obseve at at time t but moving with F (computed by modeling the signal motion in F and then tansfoming this motion to F ) is ( ) = σ tv= c t eˆ σβ = R ( t+ σ t) R () t σ tr () t (16) p whee σ=1 when modeling a signal tansmitted fom, and σ=-1 when modeling a signal eceived at. gain, is the appaent elative position of with espect to, so that the light path ange is c t and β=v/c. This fomula genealizes equation ( ) 1 to cases of tansmission and eception. The vecto p is the pope appaent elative position of with espect to, whee the tem pope indicates that this quantity is computed as peceived by (eally, by an obseve at moving in a co-moving inetial fame). When the light time delay t is small, it is possible to constuct altenate epesentations of planetay abeation that appoximate the exact expession (16). Expanding R in a Taylo seies in time, we find: 8 of 10

9 Using (17) in (16), we find: R ( t+ σ t) = R () t + σ tr () t + (17) { } R () t R () t + σ t R () t R () t (18) p which is equation ( ) 1 genealized to both tansmit and eceive cases. Similaly, expanding R in a Taylo seies in time, we find: Using (19) in (16), we find: that can be simplified to R ( t) = R ( t+ σ t) σ tr ( t+ σ t) + (19) { } R ( t+ σ t) R ( t+ σ t) + σ t R ( t+ σ t) R ( t) (20) p R ( t+ σ t) R ( t+ σ t) (21) p when the last expession in (20) is small. [This will be small fo small t and small acceleation of.] This genealizes equation ( ) 1 fo tansmit and eceive cases. The pope appaent diection is computed to be eˆ σβ pˆ = eˆ σβ that of couse agees with the value computed fo stella abeation in the case of eception. The pope appaent diection depends on t only though. lso note that the pope appaent ange p is not the same as the light path ange, no is the pope appaent ange the same in the case of tansmission and eception. This is consistent with Special Relativity as distances in F and F diffe. (22) Optical Measuements Optical obsevations of satellite position ae made by measuing the appaent satellite location against known stas in the telescope field of view. Obsevations collected in this manne can be used in detemining the obit of the satellite. These obsevations ae a function of the coections which have been applied to the sta positions. Typically these coections include such effects as the pope motion and paallax of the stas. Sta coodinate coections may optionally include annual and diunal abeation due to the motion of the obseve. Effects not accounted fo in the computation of the sta coodinates must be accounted fo sepaately in obsevation pocessing. Fo example, omission of diunal abeation fom the sta positions equies a diunal abeation coection in obit detemination. Regadless of the coections made to the sta catalog, 9 of 10

10 obit detemination must also account fo the motion of the satellite duing the time it takes fo light to tavel fom the satellite to the obseve. 1 Explanatoy Supplement to the stonomical lmanac, Ken Seidelmann, ed., of 10