Deflection of light by the field of a binary system

4.7 MB·English

by  Nahid Ahmadi

Deflection of light by the field of a binary system
Nahid Ahmadi
Department of Physics, University of Tehran, Kargar Avenue North, Tehran 14395-547, Iran
Zainab Sedaghatmanesh
Department of Physics, University of Tehran, Kargar Avenue North, Tehran 14395-547, Iran and
Department of Physics, Kharazmi University, Tehran, Iran
We examine the motion of a photon in the gravitational field of a binary system. The equations
ofmotionaregeodesicequationsinaSchwarzchildbackgroundwithatidalforce. Wespecializethe
equationstothatofanedge-onbinaryandusethemethodofosculatingelementstointegratethem.
Thisworkhelpstoidentifyabinarysystemthroughthegravitationallightdeflectionofonemember
in the gravitational field of the other member. It is found that the effects of the companion body
on a photon passing the edge of the star can be potentially detected by astrometric satellites with
5 µas precision, if the ratio of the Schwarzchid radius to the star radius, Gm ≥10−5. Two different
c2R
1 cumulative effects on the photon path are identified.
0
2
I. INTRODUCTION andtheotherscale istheradiusofthecircular orbit, de-
r
p notedbyr . Letρbetheseparationbetweenthephoton
o
A and the black hole. We assume that i) M/ρ is small and
How is the orbit of a photon that is affected by the
ii) the second body in the binary is remote. In a refer-
7 field of a binary system? This is a three- body problem
enceframethatmoveswiththemainbody, theeffectsof
2 with gravitational interactions. Adding one body to the
thesecondbodyonthegeometryofthespacetimecanbe
famoustwo-bodyproblemofaphotoninagravitational
] described by tidal potentials. These are given by a lin-
c field brings a remarkable complexity to the problem. It
earperturbationabouttheSchwarzchildsolution;sothat
q is however believed that in a complete treatment of such
theperturbedmetricsatisfiesthevacuumfieldequations,
- motions and with todays highly accurate astronomical
r perturbatively.
g observationsinhand,theseeffectsmustbetakenintoac-
[ count. This subject has been a field of interest since the
It is straightforward to apply tidal effects to a system
astrometric space mission like GAIA planned to position
2 comprising a photon and a gravitating body, but solving
v the stars and celestial objects with one microarcsecond the equations corresponding to the path of the photon
1 level of accuracy [1]. At this level, many subtle effects
in this perturbed spacetime would be tricky. A suitable
2 are potentially observable. One of them would probably
framework for solving the path based on osculating or-
9 be the deflection of light by the field of a binary system.
bitshasalreadybeensuggestedin[2]. Inthisframework,
3 When a photon moves in the (weak gravity) field of an
0 the motion of the photon is at all times described by
isolated spherical body, the leading order corrections to
. a family of null geodesics, called osculating trajectories,
1 thephoton(Newtonianpath)arescaledby Gm,whereR
c2R characterized by a set of constants of motion. A nice re-
0 istheclosestdistancetothegravitatingbodythephoton
5 formulationofthevariousaspectsoftheoriginalproblem
can reach and m is the mass of such body. Considering
1 canbeachievedbytheevolutionoftheseconstants,ifwe
the scales characterizing our problem, it is tempting to
: know the interacting forces. The evolution of this set of
v ask whether there are any relativistic effects produced
constantsisconstrainedbyasetoffirstorderdifferential
i by the second body in the binary that we should be con-
X equations, known as the osculating conditions. In this
cernedof, orthetracesleftbythecompanyingbodystill
r paper we calculate the tidal forces acting on the photon
a remains a challenge for future missions?
employing a metric calculated in a paper by Taylor and
Inthiswork,weassumethatphotonsmoveonthenull Poisson [3]. In their work, the main gravitating body
paths and are influenced by a binary system, with two was a black hole, they used this metric to find the dy-
equal masses, in a circular motion. We also assume that namics of a black hole, i.e, those produced by an influx
the radius of the binary orbit is large, compared to the ofgravitationalenergyacrossthehorizon(liketidalheat-
distance between photon and the member of the binary ing or increase in mass, angular momentum and surface
whichwecallitthemain body. Inadequatelyweakgrav- area [4]) of the black hole. Our interest in this paper is
ity regimes, effects like the deviation of the photon tra- mostly in the consequences of the tidal interactions on
jectory from straight line, may be well described by the a photon orbital dynamics. The metric given by Taylor
post-Newtonian approximation. The latter assumption andPoissonisnotnecessarilyrestrictedtotheweakfield
permits an approximate solution to our problem by re- approximation and can be equally used in small hole ap-
ducingthethree-bodyproblemtothemotionofaphoton proximation,inwhichthegravitatingbodyisassumedto
in an effective field of the orbiting bodies. be much smaller than the second body. In our problem,
Tobetterdiscussthedynamicsofthissystem,weintro- we need to know the tidal fields and their effects on the
duce two lengthscales. One scale is given by M := Gm, structure of spacetime around the gravitating body, all
c2
2
withintheweakfieldapproximationtogeneralrelativity. would like to focus on the leading effects and neglect the
Assuming that the photon path would not be too dif- smaller ones; so we find the simultaneously weak field
ferent from the Schwarzchild light paths, we will include and slow motion limit of the metric. These perturba-
the force components in the osculating equations given tions to the metric are considered as static fields on a
in [2] to obtain the observable variations at any point of flatbackground. Wethencalculatedifferentcomponents
thepath. Ourresultscanbeevaluatedinanycoordinate of the perturbing force and the equations governing the
systems; this guarantees that those are observable quan- photon path in section III. In section IV, the evolution
tities. For example, we find that the total deflection an- of constants of motion is discussed. The discussion will
gledependsonthe secondbody position andvariesfrom be concluded with the summary and conclusions of the
(cid:20) (cid:16) (cid:17)3(cid:21) results in section V.
a minimum equal to 4M 1 π R∗ to a maximum
R∗ − ro
(cid:20) (cid:16) (cid:17)3(cid:21)
4RM∗ 1+2π Rro∗ , where R∗ is the closest distance to
II. TIDAL ENVIRONMENT AROUND A
the main body that the photon could reach, if it were an
GRAVITATING BODY
isolated body.
The structure of this paper is as follows: In Section
II, we start with the metric describing the tidal envi- The general description of the spacetime around a
ronment around a gravitating body. We find the tidally spherical gravitating body in an arbitrary tidal environ-
perturbed metric in Schwarzchid coordinates by a coor- ment is given by a metric, which has the following form
dinate transformation (from harmonic coordinates). We in Schwarzchild coordinates, (cid:0)t,ρ,θA =(θ,φ)(cid:1),
(cid:18) M(cid:19) (cid:18) M(cid:19)2 (cid:18) ρ3 (cid:19)
g = 1 2 ρ2 1 2 q+O , (2.1a)
tt − − ρ − − ρ E 2
R L
g =0, (2.1b)
ρt
2 (cid:18)v ρ3 (cid:19)
g = ρ3 qΩa +O , (2.1c)
At 3 Ba A c 2
R L
(cid:34) 1 (cid:18) M(cid:19)−2(cid:35) (cid:18) ρ3 (cid:19)
g = Mρ3+ M3 1 qΩa +O , (2.1d)
ρA − 3 − ρ Ea A 2
R L
(cid:18) M(cid:19)−1 (cid:18)v ρ3 (cid:19)
g = 1 2 ρ2 q+O , (2.1e)
ρρ − ρ − E c 2
R L
(cid:34) (cid:18) (cid:19)−1(cid:18) (cid:19)2
M M
g =ρ2 Ω ρ2 1 1 2 Ω q
AB AB AB
− − ρ − ρ E
+(cid:32)1M3 (cid:18)1 M(cid:19)−1 Mρ(cid:18)1 M(cid:19)(cid:33)(cid:0)2 ΩaΩb + qΩ (cid:1)(cid:35)+O(cid:18) ρ3 (cid:19)g . (2.1f)
3 ρ − ρ − − ρ Eab A B E AB 2 ab
R L
Here M is gravitational radius of our body, Ωa = of a black hole placed within a post-Newtonian external
A
∂Ωa which satisfies ΩABΩaΩb = γab and ΩAΩa = spacetime. Appendix A presents some crucial results of
∂θA A B a
0. Furthermore, ΩA = ΩABΩ , Ωa := xa = that paper. Our work is a continuation of their effort,
a aB r and this coordinate transformation brings the metric of
(sinθcosφ,sinθsinφ,cosθ) is a unit radial vector, Ω
AB
isthemetricontheunittwo-sphere,xa’sarecoordinates the distorted black hole (A.1) to the new form given in
(2.1). This metric applies to the spacetime outside any
of a Cartesian system, and γ = δ Ω Ω is a pro-
ab ab a b
jection tensor on the direction orthogo−nal to Ωa. The spherical body immersed in a tidal environment. It is
linearized about the Schwarzchild solution and satisfies
domain of validity of the metric is the spherical body
(approximate) vacuum field equations. The tidal envi-
neighborhood. To find this metric, we used the results
ronment is presented as a functional of arbitrary exter-
of [3], and performed a coordinate transformation from
theharmonicscoordinates(cid:0)x0,xa(cid:1)to(t,ρ,θ,φ),givenby nalpotentials{Eq,Eaq,Eaqb,Baq}builtfromthelowestorder
x0 = ct,xa = rΩa(cid:0)θA(cid:1), r = ρ−M. In [3], Taylor and tthideaml metormiceisntasc,cEuarbataentdoBalalbo.rWdeirtshiinnMthiasnadptphreoxniemglaetcitoend,
Poisson, examined the motion and the tidal dynamics ρ
termsinvolvethirdorderin ρ thatcomefromhigheror-
ro
3
der tidal moments or the time derivative of quadrapole 1. Althoughthegravitomagnetictidal(vector)poten-
moments. Although the potentials that characterize the tials exist in a relativistic description of the geom-
tidal perturbations are completely general, in this work etry around a gravitating body, they do not ap-
we require that the gravitating body be a member of a pear in weak field limit and slow motion context
binary system. (v/c 1,wherev isthetidalenvironmentvelocity
(cid:28)
In studying the path of a photon through a curved ge- scale). This is also true for gravito-electric vector
ometry, it is a well known practice to consider the met- and tensor potentials.
ric perturbations as fields defined on a flat background
spacetime. Inthiswaythetidalenvironmentisdescribed 2. ThesphericalsymmetryoftheSchwarzchildgeom-
bytheweakfieldlimitofthemetric. Followingthesame etryisperturbationallybroken;sotheBirkhoffthe-
practice, we take M as well as the potential terms pro- oremwillletustohaveanon-staticvacuumspace-
ρ
duced by the second body, i.e the linear perturbation time.
termstoflatMinkowskispace,tobesmall. Fortheprob-
lemunderstudy,theradiusofthebinaryislargeandthe 3. The tidal geometry is described by a scalar poten-
termsbeyondthequadrapoletidalmomentswerealready tial and each scalar potential transforms as such
assumed to be small1. For this procedure to make sense, under parity. This ensures that the deformed met-
(cid:113) ric be conserved under a parity transformation.
weshouldinvoketheslowmotionlimit, v = M 1as
c ro (cid:28)
well. We introduce two dimensionless parameters which 4. If we expand the tidal potential given in (2.4), in
are almost equally small and denote them by (cid:15) terms of spherical harmonics we will have
(cid:18) (cid:19)4 (cid:114)
(cid:15):= M ρ . (2.2) q = πA(cid:104) Y0+√24(cid:0)Y2+Y−2(cid:1)+O((cid:15))(cid:105). (2.5)
ρ ≈ ro E 5 − 2 2 2
In this limit, the nonzero components of the metric are It is observed that the coefficients of Y±1 are zero,
2
thus written as and the metric components are invariant under
θ π θ. This fact demonstrates an additional
gtt = (cid:18)1 2M +ρ2 q(cid:19)+O(cid:0)(cid:15)2(cid:1), (2.3a) sy→mmet−ry with respect to the plane θ = π2. The
− − ρ E presence of the second body has greatly reduced
gρρ =1+2M ρ2 q+O(cid:0)(cid:15)2(cid:1), (2.3b) tthoe(disiosmcreettrey) (sLuibeg)rgoruopusp, nofamScehlwyaprazrcihtyildasnpdacmeitrirmoer
ρ − E
g =ρ2Ω (cid:2)1 ρ2 q+O(cid:0)(cid:15)2(cid:1)(cid:3), (2.3c) symmetry.
AB AB
− E
where q = 1 ΩaΩb and is a symmetric-trace free
E c2Eab Eab III. NULL TRAJECTORIES IN THE
tensor,whichitscomponentsforcircularorbitmotionare
PERTURBED SPACETIME
calculated by matching the local metric with the global
metricthatincludethegravitatingbodyandtheexternal
Thefirststepbeforewritingthenullgeodesicequations
spacetime [3]. These components are given in Appendix.
By substituting (A.3) in (A.2), the scalar potential, q, istocalculatetheChristoffelsymbolsforthismetric. The
E perturbed geodesic equations will then be written as
is obtained as
q =A(cid:2)1 3cos2θ+3sin2θcos2(φ ψ)+O((cid:15))(cid:3), d2xµ +Γ¯µ dxν dxσ =Fµ, (3.1)
E − − (2.4) dλ2 νσ dλ dλ
where A := M is a dimensionful parameter and ψ =
−2ro3 where
ωt, where ω denotes the angular frequency of the tidal
momentswithrespecttotheframemovingwiththemain dxν dxσ
Fµ = δΓµ , Γµ =Γ¯µ +δΓµ . (3.2)
body. Beforeexploringthebehaviorofthephotoninthis − νσ dλ dλ νσ νσ νσ
spacetime, there are some points that we would like to
add: Here Γ¯s and δΓs are components of the Schwarzchild
Christoffelsymbolsandtheirperturbations,respectively.
Appendix B gives the full list of these symbols in the
weak field limit. The right hand side of (3.1) represents
1 The light defection in the post-linear gravitational field of two the perturbing force components.
boundedmassesforthecasewhentheimpactparameterismuch It does not seem to be much hope for solving the set
larger(5timesormore)thanthedistancebetweenbinarymem-
of equations (3.1) in a general case. Fortunately, there
bers,iscalculatedin[5]. Inthefollowing,weshowthatthelinear
aresomespecialcasesinwhichtheremainingsymmetries
termsaresosmallthatwecanconfidentlyneglectthepost-linear
terms. simplify the task. It is easy to see that the plane θ = π
2
4
satisfies the equation for θ(λ).2 Therefore, if we special- with parameter λ, is chosen as xµ. We recall that in
G
izetothecaseinwhichphotonsmoveinequatorialplane weak field approximation, the relations describing forced
of the binary system, i.e, if the inclination parameter, i, Schwarzchild null geodesics are
is exactly π/2 and the binary orbit is edge-on, the di-
sin(φ ∆(φ))
rection of angular momentum will be conserved and the ρ−1(φ)= −
R(φ)
photon will not leave the plane, because Fθ = 0. Other
(cid:20) (cid:21)
components of the tidal force applied to the photon will 3M 1
+ 1+ cos2(φ ∆(φ)) , (3.5a)
then be 2R2(φ) 3 −
Ft = 1ρ2(cid:16)ρ2φ˙2+ρ˙2 t˙2(cid:17)∂ q φ(λ)=∆(λ)+(cid:90) λL(cid:16)λ˜(cid:17)ρ−2(cid:16)λ˜(cid:17)dλ˜, (3.5b)
t
2 − E 0
−2ρt˙ρ˙Eq−ρ2t˙φ˙∂φEq+O(cid:0)(cid:15)2(cid:1) (3.3a) (cid:90) λ E(cid:16)λ˜(cid:17)
Fρ =ρ(cid:16)ρ˙2−t˙2−ρ2φ˙2(cid:17)Eq t(λ)=T (λ)+ 0 1 2Mρ−1(cid:16)λ˜(cid:17)dλ˜, (3.5c)
+1ρ(cid:16)2(cid:0)t˙∂tEq+ρ˙∂φE(cid:17)q(cid:1)+O(cid:0)(cid:15)2(cid:1) (3.3b) ρ˙2 =E2 L2 (cid:18)1 −2M(cid:19). (3.5d)
Fφ = ρ2φ˙2 ρ˙2 t˙2 ∂ q − ρ2 − ρ
φ
2 − − E
+ρ2∂ q+2ρρ˙φ˙ q+O(cid:0)(cid:15)2(cid:1). (3.3c) Here dot denotes derivative with respect to λ and
tE E the osculating elements on these trajectories are
E,L,R,∆,T . Someosculatingelements,likeR,T and
Even with above simplifications, integrating the equa- { }
∆, that characterize the initial position on the fiducial
tions of motion and determining the photon path will
trajectory, are usually referred to as the positional ele-
not be simple. Although the perturbing forces are com-
ments; these elements typically change whenever the ge-
plicated, we believe that these forces keep the form of
ometry of spacetime is affected with a new interaction.
the photon path. Therefore, the leading correction to
Other elements, like L and E, which are called princi-
the Schwarzchild null curves can be determined by the
pal elements, determine on which trajectory the photon
method of Variation of Parameters (VoP). As described
is moving. We can ignore T and evolve t explicitly by
in [2], in this method one employs a known solution of
using
a fiducial system of differential equations as an ansatz
for solving the system under study. The known solution, E ρ2
xµG, playstheroleofinstantaneouslytangential(oroscu- t´= L1 2M , (3.6)
lating) trajectories to the true path, xµ(λ). Each point − ρ
on the truepath with parameter λ is coincides geometri- whichisequivalenttotheevolvingoft T,directly. The
−
callyanddynamicallywithanosculatingtrajectorywith flow lines that satisfy (3.4) are given by
coonnestsapnetcsifiiesdesqeutaolftocotnhsetannutsm,bIeαr; otfhephnausmebsepracoef tinhdesee- E´ = R2 1 Ft+O(cid:0)(cid:15)2(cid:1), (3.7a)
L sin2(φ ∆)
pendent variables. The constants are furnished with λ −
dependence and evolve along a flow line on a cross sec- ∆´ = R3 1 Fρ+O(cid:0)(cid:15)2(cid:1), (3.7b)
tionofphasespacethatisconsistentwithnullcondition. −L2 sin(φ ∆)
−
Ecoancdhitainondseveryflowlineisconstrainedbytheosculating L´ = R4 1 Fφ+O(cid:0)(cid:15)2(cid:1), (3.7c)
L sin4(φ ∆)
∂x˙µGI˙α =Fµ, ∂xµGI˙α =0, (3.4) R´ = R4 cos(φ−−∆)Fρ+O(cid:0)(cid:15)2(cid:1), (3.7d)
∂Iα ∂Iα L2 sin2(φ ∆)
−
but different flow lines may correspond to very different t´= E sin(φ ∆)[sin(φ ∆)+4M]+O(cid:0)(cid:15)2(cid:1).
interactions. This reformulation of the problem provides R2L − −
an insight into the effects of perturbing forces on various (3.7e)
aspects of the motion. In the framework of VoP, the
In these equations, φ has been used instead of λ (or t),
perturbing forces are not assumed to be small and the
as an independent variable and prime denotes derivative
onlyrestrictionisthattheforcesmustpreservetheshape
withrespecttoφ. Theequationsgoverningtheevolution
of the path.
of E and L could have been calculated by the covariant
Following [2], these constants are called osculating
formulation of osculating conditions [6].
elements, and the Schwarzchild null geodesics family,
The variations in the osculating elements are assumed
tobesmall,thereforeonecanachieveagoodapproxima-
tion of these elements by substituting the leading order
expressions,
2 If we calculate (3.1) for xµ = θ, different terms include either
Γθθµθ˙µ˙ orΓθµν whichcanbeeasilyverifiedfrom(B.1)tobezero ρ= R∗ (1+O((cid:15))), (3.8a)
atθ=π/2. sin(φ ∆ )
∗
−
5
t˙=E (1+O((cid:15))), (3.8b) intotherighthandsideofequations(3.7). Herethesetof
∗
(cid:18)sin2(φ ∆ ) (cid:19) constantsthatthephotontrajectorycouldhaveassumed,
ρ˙2 =E2 L2 − ∗ +O((cid:15)) , (3.8c) ifthemainbodywereinacompleteisolation,aredenoted
∗ − ∗ R2
∗ by E ,L ,R ,∆ . The different components of the
∗ ∗ ∗ ∗
φ˙ = L∗ sin2(φ ∆ )+O((cid:15)), (3.8d) pert{urbing force in}the plane θ =π/2 has the form of
R2 − ∗
∗
(cid:20) (cid:21)
Ft =2E L A 1 cot(φ ∆ )[1+3cos2(φ ψ)]+3sin2(φ ψ) +O(cid:0)(cid:15)2(cid:1), (3.9a)
∗ ∗ ∗
k − − −
Fρ = 2E L A(cid:20)kcos(φ ∆ )[1+3cos2(φ ψ)]+3cos(φ ∆ )cos2(φ ψ)(cid:20) E∗R∗2ω L (cid:21)(cid:21)+O(cid:0)(cid:15)2(cid:1),
− ∗ ∗ − ∗ − − ∗ − sin2(φ ∆ ) − ∗
∗
−
(3.9b)
(cid:20) (cid:20) (cid:21)(cid:21)
Fφ =2E2A k[1+3cos2(φ ψ)]sin2(φ ∆ )+3sin2(φ ψ) 1+ L∗ω k2sin2(φ ∆ ) +O(cid:0)(cid:15)2(cid:1). (3.9c)
∗ − − − ∗ − E − − ∗
∗
Here k :=E R /L is a dimensionless parameter, which the perturbing force can be written as
∗ ∗ ∗
its value can be found by substituting, ρ=R in (3.5d),
∗ (cid:20) (cid:21)
k =1+O((cid:15)). In the derivation of the force components, Ft =2E L Acot(φ ∆ ) 3cos(φ−2ψ+∆∗) +1 ,
∗ ∗ ∗
wehaveusedthesquarerootof(3.8c)withaminussign, − cos(φ ∆ )
∗
−
ρ˙ = E cos(φ ∆ )+O((cid:15)). (3.11a)
∗ ∗
− −
We recall that in preceding equations, ψ is a function (cid:20) sin(φ 2ψ+∆ ) (cid:21)
of t; so formally, its explicit dependence on φ must be Fρ =2E∗L∗Asin(φ ∆∗) 3 − ∗ 1 ,
− sin(φ ∆ ) −
∗
considered. However, these equations will turn out to be −
(3.11b)
simpler by recognizing that in the problem under study,
(cid:20) (cid:21)
sin(φ 2ψ+∆ )
osculating elements undergo two types of changes. The Fφ =2E2Acos(φ ∆ ) 3 − ∗ 1 .
first is an oscillation with a period equal to the period of ∗ − ∗ sin(φ ∆∗) −
−
tidal moments (or multiple of it); the second is a steady (3.11c)
drift in φ. In other words, we have two timescales; t
and ω−1. On short timescale, t M, the photon moves The evolution of the conserved quantities will then be
on a geodesy of the Schwarzchild≈spacetime with a static reproduced by substituting the preceding equations into
tidal field, characterized by E,L,R,∆ . Over longer (3.7) as
(cid:113) { }
timescale, ω−1 M M , the displacement of the (cid:20) (cid:21)
≈ ro3 ≈ (cid:15)185 ∆´ = 2AR2 3sin(φ−2ψ+∆∗) 1 , (3.12a)
second body changes ψ and causes the path to evolve. − ∗ sin(φ ∆ ) −
∗
It is clearly desirable to compute an accurate trajectory −
R´ (cid:20) sin(φ 2ψ+∆ ) (cid:21)
for the whole domain of the validity of φ (or t). How- =2AR2cot(φ ∆ ) 3 − ∗ 1 ,
ever, the linear perturbation method is limited to pro- R∗ ∗ − ∗ sin(φ ∆∗) −
−
ducing a model for the trajectory over rapid timescale, (3.12b)
(oransnapshotofthetrajectory,)inwhichthetrajectory E´ cos(φ ∆ ) (cid:20) cos(φ 2ψ+∆ ) (cid:21)
is described by the deviations from the pivot elements, =2AR2 − ∗ 3 − ∗ +1 ,
E∗,L∗,R∗,∆∗ , andthenewconfigurations(ofthesec- E∗ ∗sin3(φ−∆∗) cos(φ−∆∗)
{ } (3.12c)
ond body) are neglected. Such approximations fall out
of phase with the true path after the longer timescale L´ cos(φ ∆ ) (cid:20) sin(φ 2ψ+∆ ) (cid:21)
c≈an(cid:15)−b81e5Mpr.ovIindefdacbtyaaritgworoo-tuismpesrceascleripatniaolnysfiosr, tbhuet pcaotnh- L∗ =2AR∗2sin3(φ−−∆∗∗) 3 sin(−φ−∆∗) ∗ −(31.12.d)
sidering the order of accuracy by which the short term
secular changes are captured, at the cost of discarding It is easily seen that the corrections imposed to the
long term effects, it is sensible to assume that ψ˙ φ˙ or Schwarzchild null paths by the second body are of or-
(cid:28) (cid:16) (cid:17)
ωt˙ R2E ω 1 der AR∗2, that is O (cid:15)74 . These corrections are so
∗ ∗ 1. (3.10) small that, even for photons with large impact param-
φ˙ ≈ L∗ sin2(φ ∆∗) (cid:28) eters, we can neglect higher order terms in (2.3) with
−
confidence. These equations appear to be singular at
After invoking this approximation, the components of φ=∆ ,π+∆ . The singularities correspond to the fact
∗ ∗
6
that Schwarzchild geodesics loose their geometric mean-
ing for asymptotic values of φ.
In the following section, we will give expressions for
osculating elements; once these functions are known, the
path is obtained from equations (3.5). Considering the
long separation of the two timescales, we assumed that 2200
the geometry around the main body has time transla-
tion symmetry. Before concluding this section, we are
1100
interested in using this symmetry to calculate energy of ΔΔ
the perturbed system, E¯, as a constant on the path.
00
The Killing vector whose conserved quantity is energy,
is K = ∂ . The conserved energy is then given by
∂t −−1100
ππ
E¯ = p Kµ =t˙(cid:18)1 2M +ρ2 q+O(cid:0)(cid:15)2(cid:1)(cid:19) (3.13) ππ44 88
− µ − ρ E 3344ππ 3388ππ
This is a perturbationally constant quantity even for the ψψ 5544ππ 5588ππ φφ
paths outside of the equatorial plane. Moreover, for the
trajectories constrained to the equatorial plane, the di-
rection of angular momentum is also conserved. FIG.1: Variationsof∆−∆∗vs. φandψ. Here∆∗isassumed
to be zero. On the vertical axis, we have chosen AR2 =1
∗
IV. EVOLUTION OF OSCULATING
CONSTANTS
The expression for the evolution of ∆ in (3.12a) can
now be used to obtain the explicit dependence on ψ at
each φ. This gives
1155
∆(φ)=∆ +2AR2[φ(1 3cos2(ψ ∆ )) 1100
∗ ∗ − − ∗ 55
+ 3sin2(ψ ∆ )ln(sin ∆ φ)]. (4.1) //RR**--11
− ∗ | ∗− | 00
55
Fig. 1 shows the variations of ∆ at different points of −−
1100
thepathfordifferentψ directionsattheinitialcondition −−
1155
∆∗ = 0. The contribution of the second body in the −− ππ88
ππ
total deflection angle of a light ray passing near a binary 44
system is 3344ππ 3388ππ
(cid:90) ∆∗+π ψψ 5544ππ 5588ππ φφ
∆´dφ=2πAR2[1 3cos2(ψ ∆ )]. (4.2)
∗ − − ∗
∆∗
This implies that for edge-on binaries, the second body FIG. 2: Variations of R/R∗−1 vs. φ and ψ. On the vertical
gives maximum increase in the deflection angle when axis, we have chosen AR∗2 =1
ψ ∆ = (cid:8)π,3π(cid:9) and maximum decrease is obtained
− ∗ 2 2
wbihtsenwiψth−a∆rb∗it=ra{ry0,iπnc}l.inIant[i7o]n,,itthiesmshaoxwimnathlpatosasmibolengligohrt- − (1−3cos2(∆∗−ψ))lnsin2(φ−∆∗)(cid:3). (4.4)
deflectionisattainedforedge-onbinaries,(i=π/2). Our This amounts to a total change in R equal to
calculations complement the criteria for orbital elements
by using those the maximal changes in positional ele- 12πAR∗3sin2(∆∗−ψ). (4.5)
ments are determined. The various gravitational effects
The photon gets a maximum value of total R- deflection
in the light propagation are additive in weak field limit; when ψ ∆ = (cid:8)π,3π,5π,7π(cid:9); no change in total R is
so for the total deflection angle, δ, we will find that expected−wh∗en ψ 4 ∆4 =4(cid:8)π4,π,3π(cid:9).
− ∗ 2 2
M M Obviously, one can find a different parametrization of
4R −4πAR∗2 ≤δ ≤4R +8πAR∗2. (4.3) the path by first integrating (3.7e) and keeping the lead-
∗ ∗
ing order terms to produce,
By integrating (3.12b), one can obtain (cid:20) (cid:21)
E 1 1
∗
t(φ)= φ sin2(φ ∆ )+4Mcos(φ ∆ ) ,
R L R2 2 − 4 − ∗ − ∗
=1+2AR2[6sin2(∆ ψ)(1 cot(φ ∆ )) ∗ ∗
R ∗ ∗− − − ∗ (4.6)
∗
7
4400
1100 3300
2200
00
1100
//EE**--11 //LL**--11
1100
−− 00
−−2200 −−1100
2200
3300 −−
−− ππ ππ
88 88
ππ ππ
44 44
3344ππ 3388ππ 3344ππ 3388ππ
ψψ 5544ππ 5588ππ φφ ψψ 5544ππ 5588ππ φφ
FIG. 3: Variations of E/E −1 vs. φ and ψ. On the vertical FIG. 4: Variations of L/L −1 vs. φ and ψ. On the vertical
∗ ∗
axis, we have chosen AR2 =1 axis, we have chosen AR2 =1
∗ ∗
V. SUMMARY AND CONCLUSIONS
and then omitting φ in favor of t, numerically. The ex-
pressions describing the evolution of other osculating el-
In this paper, we considered a photon in the gravi-
ements, up to the leading order in (cid:15), are
tational field of a binary system and found the form of
trajectories invoking the weak field and slow motion ap-
E proximations. In practice, we found the magnitude of
E =1+AR∗2[6sin2(∆∗−ψ)cot(φ−∆∗) secular changes to the path of a light ray propagating in
∗
(cid:21) the field of a spherical gravitating body, that arises in
1+3cos2(∆ ψ)
∗− , (4.7a) the presence of a companion body. This work illustrates
− sin2(φ ∆∗) the application of the osculating elements formalism to
−
L =1+AR2(cid:2) 6sin2(∆ ψ)cot3(φ ∆ ) the computation of photons paths evolutions in the the
L ∗ − ∗− − ∗ Schwarzchild spacetime, as well as the tidal geometry
∗
1 3cos2(∆ ψ)(cid:21) aroundablackholeinreducingaphoton-binaryproblem
∗
+ − − . (4.7b) to an effective two body problem. This work shows how
sin2(φ ∆ )
− ∗ a companion star can affect the curvature of a light ray
broughtaboutbythemainlens. Itisknownthatedge-on
binaries deflect the light more than others [7]; this cri-
InFigs.1-4,weshowtheevolutionofthepathunderthe
terion is further developed in this paper. Provided that
influenceofthesecondbodyandinitialcondition∆ =0.
∗
the companion body is in a right distance and at a right
To avoid the singularities at asymptotes, φ=0,π, the φ
orbital phase, in principle, it can be traced with a sensi-
variationsdoesnotcoverthewholedomain,alsowehave
chosen AR∗2 = 1. The four figures show the two posi- tairveeddeetteercmtoinr.edTbhyetrhigehgtedoimsteatnrcyeo,fros,paacnedtipmheasaer,oψun−d∆th∗e,
tional elements, (∆,R), and the two principal elements,
main lens. Equation (3.12) indicates that edge-on bina-
(E,L), as functions of φ and ψ. The comparison of be-
havior of these elements at asymptotes shows that there ries with ro = O(cid:16)(cid:15)−54M(cid:17), and ψ −∆∗ = (cid:8)π2,32π(cid:9) can
is a secular change in (∆,R); but there is no changes in (cid:16) (cid:17)
the values of the principal elements in these limits. For produce deflections of order O (cid:15)74 . Therefore, these
particles moving on timelike bound orbits, it is believed corrections have potentially detectable effects on GAIA
that the changes in principal elements occur when per- (or any other µas 10−8 astrometric mission) observ-
turbing forces include a dissipative part [8]. Likewise, ables, if (cid:15) 10−5. ≈This condition holds for known typi-
≥
equations(4.7b)and(4.7a)implythatL(0)=L(π)and calbinarysystems. Inthisestimation,itisassumedthat
E(0)=E(π) for massless particles; even though the to- the photon passes by the edge of the star.
tal angular momentum is not conserved in our problem. The importance of determining such corrections to
Although this result is obtained for a tidal interaction Schwarzchild null geodesics can be understood when as-
andedge-onbinaries,itseemstobealsovalidforgeneral trometricinformationisreadoffthelightpathsatahigh
inclinations. precisionlevel. TheastrometricobservationsofaGAIA-
8
like satelite has been modeled within the PPN formal- curves. Such models provide a convenient framework for
ism to Post-Newtonian gravity. The sensitivity of such mutual verification of the results in an astrometric field
astronomy to some PPN parameters [11] as well as the common in different sky missions.
light bending due to quadrapole moment of an axisym-
metric planet (like Jupiter) [12] has been established. In
such feasibility studies, one must consider the light de-
Appendix A TIDAL ENVIRONMENT AROUND
flection in the vicinity of a star which is i) a member
THE MAIN BODY
of a binary system and ii) crosses one of the astromet-
ric fields during the GAIA’s mission. A comparision be-
tween the details of the background field corresponding In this appendix we list some results presented in [3]
to the time when the star is at the center of the field which investigated a tidally deformed spacetime around
with that when it is not, will help to extract the contri- a nonrotating black hole. The tidal environment around
bution of the binary system. The difference between the a slowly rotating black hole has been studied recently
two patterns may then be fitted to the deflection model by Poisson [9]. To capture the effects associated with
to determine whether the effect is detectable or not. In the dragging of inertial frames, this calculation required
spite of the common belief that 50 percent of stars are to go beyond first order perturbation theory. When the
a member of a binary (or multiple) system [13] and the rotational deformations are applied to a circular binary,
order of correction terms is in GAIA’s accuracy level, the tidal moments will incorporate terms that involve
finding a favourable field and binary system with known powers of v/c. Since these two metrics are the same at
ephemerides is a big challenge that is out of the scope of theordersthatwerequire,thehigherordertermsdonot
this paper. On the other hand, the trace of wide binary affect on the results of our work. Some notations we use
lensesstudiedinthispapermaybefoundinmicrolensing inourwork,andareadaptedfrom[3],willbeintroduced
light curves. If repeating microlensing peaks [14] appear here. Wewillexpresstheresultsintheformrequiredfor
in the microlensing light curves during the GAIA’s mis- our specific purpose.
sionperiod,itwillbeagoodsignalforabinarycandidate The metric components of the black hole immersed in
with M r . Analytical results derived in this paper an arbitrary tidal environment, in harmonic coordinates
o
can be u(cid:28)sed in models fitted to the microlensing light (cid:0)x0,xa(cid:1), are given by
1 M (cid:18) M(cid:19)2 (cid:18) r3 (cid:19)
g = − r r2 1 q+O , (A.1a)
00 −1+ M − − r E 2
r R L
2 (cid:18) M(cid:19)(cid:18) M(cid:19)2 (cid:18)v r3 (cid:19)
g = r2 1 1+ q +O , (A.1b)
0a 3 − r r Ba c 2
R L
1+ M (cid:18) M(cid:19)2 (cid:18) M(cid:19)2 (cid:18) M(cid:19)2(cid:18) M2(cid:19)2
g = r Ω Ω + 1+ γ r2 1+ qΩ Ω Mr 1+ 1+ (Ω q+ qΩ )
ab 1 M a b r ab− r E a b− r 3r2 aEb Ea b
− r
(cid:18) M(cid:19)2(cid:18) M(cid:19)3 (cid:18) M(cid:19)2(cid:18) M2(cid:19) (cid:18) r3 (cid:19)
r2 1 1+ γ q Mr 1+ 1 q +O . (A.1c)
− − r r abE − r − 3r2 Eab 2
R L
HereM isthegravitationalradiusoftheblackhole,when quadrapole moments, and by
ab ab
it is in complete isolation, xa are a system of Cartesian E B
cvoecotrodrinwahteicsh, Ωsaat:i=sfiexsraδ=Ω(sainΩθbc=os1φa,nsidnγθsin=φδ,cosθΩ) isΩa Eq = c12EabΩaΩb, (A.2a)
ab ab ab a b
is a projection tensor on the direction orthogonal−to Ωa. q = 1 γ c Ωd, (A.2b)
The contribution of the tidal environment is expressed Ea c2 a Ecd
as an expansion in the strength of the tidal coupling. q = 1 (cid:0)2γ cγ d +γ q(cid:1), (A.2c)
The expansion parameter is the inverse of the radius of Eab c2 a b Ecd abE
cbularcvkathuorlee, mRo−v1e,s.ofInthteheexnteegrnleacltesdpatceertmims,e iniswthhiechsctahlee Baq = c13(cid:15)apqΩpBcqΩc. (A.2d)
L
of spatial inhomogeneity in the external spacetime. At
The tidal moments are formally defined in terms of the
the lowest order in −2, the metric can be described by
a set of external poRtentials q, q, q , q built from Weyl tensor of the external spacetime. In practice,
{E Ea Eab Ba} however, for every application those are determined by
matching the local metric to the global metric that in-
9
cludes the black hole and the external spacetime. For = = 3Mvsinψ+O(cid:0)c−2(cid:1), (A.3e)
the specialized case that the black hole is part of a bi- B23 B31 − r3
o
nary system in circular motion and the global spacetime = = 3Mvcosψ+O(cid:0)c−2(cid:1). (A.3f)
is expressed in the post-Newtonian gravity regime, such B13 B32 − r3
o
amatchingwerecarriedoutin[3]and[10]. Fortheprob-
lemunderstudyinourpaperwherethetwogravitational
Here ψ is the phase related to the angular velocity, ω,
bodies in the binary system have equal masses and the
of the tidal moments with respect to the frame moving
relative orbi(cid:113)tal velocity,(cid:16)v,(cid:17)within the binary system is with the black hole, ψ = ωt = (cid:113)2Mt(cid:16)1+O(cid:0)v(cid:1)2(cid:17). It
slow, v = 2M = O (cid:15)58 , the nonvanishing compo- ro3 c
c ro can also be defined in terms of the orbital velocity of the
nents of the tidal moments are given by
secondbody,ω¯,andtheprecessionalangularfrequencyof
M (cid:20) (cid:16)v(cid:17)2(cid:21) theblackholeframerelativetothebarycentricframe,Ω,
= 1+3cos2ψ+O , (A.3a) by ψ =ω¯t¯ Ωt, in which t¯is the global time coordinate.
E11 −2r3 c −
o
M (cid:20) (cid:16)v(cid:17)2(cid:21)
= = 1+3sin2ψ+O , (A.3b)
E12 E21 −2r3 c
o
M (cid:20) (cid:16)v(cid:17)2(cid:21) Appendix B CHRISTOFFEL SYMBOLS
= 1 3cos2ψ+O , (A.3c)
E22 −2r3 − c
o
The nonvanishing Christoffel symbols built from met-
M (cid:16)v(cid:17)2
= + = +O , (A.3d) ric (2.3) are
E33 E11 E22 −r3 c
o
ΓΓΓttttttρ===−Mρ12Γρtρ+2ρρ∂=Eq12,qρ,2∂∂tEq, ΓΓΓρρρtρtt===−12ρΓ21ρρρρ∂∂t2=Eq∂Mρ,2 +q,ρEq, ΓΓΓθθθθθttρt===−−ρ1Γ12−θρρρ2ρ=∂E∂tqE,12qρ,∂∂θEq, ΓΓΓφtφφφφtρt===−−ρ1Γ21−φρρρ2ρ=∂E∂tqE2,qsi,1n2θ∂∂φEq,
ΓtAtAB =2−12∂ρθ4AΩEAB∂∂tEq, ΓρAρAB =−ρ32ΩA∂Bθ∂A∂tEEq, ΓΓθθθφAφ ==−−12siρn2θ∂cθ∂AosEθq,+ 12ρ2sinθ∂∂θEq, ΓΓφθφφθφ==12−ρ122ρs2in∂∂2φθE∂q∂φ.Eq,
(B.1)
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