Erratum: Radial infall of two compact objects:2.5-post-Newtonian linear momentum flux and associated recoil

[Phys. Rev. D 85, 104046 (2012)]

Chandra Kant Mishra(Received 5 March 2013; published 14 March 2013)

DOI: 10.1103/PhysRevD.87.069907 PACS numbers: 04.25.Nx, 04.30.Db, 97.60.Jd, 97.60.Lf, 99.10.Cd

Recently we realized that we committed a conceptual error in the argument presented in the paper to demonstrate theindependence of physical results on the arbitrary length scale r0. From Eqs. (2.6a)–(2.6c), it can be seen that although theuse of harmonic coordinates removes most of the r0 dependence from Eqs. (2.5a)–(2.5c), it introduces a new dependenceon the distance of the source (r). While computing the hereditary part of the linear momentum flux, this r was replaced bythe separation z [see, for instance Eqs. (4.3), (4.4), and (4.8)]. This is equivalent to the choice of the retarded time,u ¼ t� r=cþ ð2 Gm=c3Þ log ðz=rÞ, in contrast to the choice u ¼ t� r=c as claimed in the paper. This choice involvingzðtÞ is not suitable to make the PN expansion, c ! 1 along u ¼ constant. We find that this issue is resolved if retarded timeis chosen as u0 ¼ t� r=cþ ð2 Gm=c3Þ log ðGm=c2rÞ (similar to Ref. [1]). As a consequence of this redefinition of theretarded time, coefficients of some terms in the results change and must be corrected. The modifications resulting fromthese changes are summarized below.

(i) Equations (2.6) and (2.10) u ! u0 and r ! Gm=c2.(ii) Equations (4.3), (4.4), and (4.8) u ! u0 and z ! Gm=c2.(iii) Equations (4.5), (4.7), and (4.11) log ð8�Þ ! 3 log ð2�Þ.(iv) Equations (4.9), (4.10), and (4.12) log

�23�

�! log

�23

�þ 3 log ð�Þ.

(v) Equation (5.6) log

�23�f

�! log

�23

�þ 3 log ð�fÞ.

The numerical estimates for the recoil velocity also change significantly as compared to the ones presented in thepublished version of the work due to these modifications. For instance, for a binary with symmetric mass ratio of � ¼ 0:2,the recoil velocity accumulated until the two objects under radial infall reach the separation of 8:33 Gm=c2 (� ¼ 0:12) isof the order of 0:11 km s�1 as compared to the corresponding older estimate of 0:011 km s�1. Note that here we propose to

0.01 0.05 0.1 0.20.24ν

10-4

10-3

10-2

10-1

100

|Vki

ck| (

in k

m-s

-1)

γi=0.0γi=0.01γi=0.02γi=0.05

γf=0.12; 2.5PN Model

10 15 20 50 100

γf-1

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

|Vki

ck|(

in k

m-s

-1)

γi=0.0γi=0.01γi=0.02γi=0.05

ν=0.2; 2.5 PN Model

FIG. 1 (color online). Recoil velocity as a function of the mass parameter � (left panel) and as a function of the post-Newtonianparameter �f (right panel) has been plotted. The parameter, �, is known as symmetric mass ratio of the binary; the parameters �f ¼ðGm=c2zfÞ and �i ¼ ðGm=c2ziÞ are the post-Newtonian parameters characterizing the final and initial separation of the two objects,

respectively. For the plot in the left panel, the value of the parameter �f has been fixed to 0.12, which corresponds to the final

separation of roughly about 8:33 Gm=c2 between the two objects and then the recoil velocity as a function of the parameter � has beenplotted. Similarly, for the right panel, the value of the parameter � has been fixed to 0.2 and recoil velocity as a function of theparameter �f has been shown. These plots (both in the left and the right panel) also compare recoil velocity estimates for four different

situations based on the binary’s initial separation: �i ¼ 0:01, 0.02, 0.05, and 0.0, which correspond to the initial separation of the twoobjects of 100 Gm=c2, 50 Gm=c2, 20 Gm=c2, and 1 (infinite initial separation case), respectively.

PHYSICAL REVIEW D 87, 069907(E) (2013)

1550-7998=2013=87(6)=069907(2) 069907-1 � 2013 American Physical Society

terminate the numerical integrals when � ¼ 0:12, unlike the published version where we terminate the relevant integralswhen � ¼ 0:2. The reason is that the modified PN expression for the recoil velocity is no longer valid beyond � ¼ 0:12,and hence one has to stop at this separation. This is also a reason why the replacement of both Figs. 1 and 2 is needed apartfrom the fact that the modifications discussed above change the numerical estimates for the recoil velocity.

[1] L. E. Simone, E. Poisson, and C.M. Will, Phys. Rev. D 52, 4481 (1995).

0.01ν

10-6

10-5

10-4

10-3

10-2

10-1

100

|Vki

ck|(

in k

m-s

-1)

1PN2PN2.5PN

γf=0.12; γi=0.0

0.010.05 0.1 0.20.24 0.05 0.1 0.20.24ν

10-6

10-5

10-4

10-3

10-2

10-1

100

|Vki

ck|(

in k

m-s

-1)

Newtonian1PN1.5PN2PN2.5PN

γf=0.12; γi=0.01

0.01ν

10-6

10-5

10-4

10-3

10-2

10-1

100

|Vki

ck|(

in k

m-s

-1)

Newtonian1PN1.5PN2PN2.5PN

γf=0.12; γi=0.02

0.010.05 0.1 0.20.24 0.05 0.1 0.20.24ν

10-6

10-5

10-4

10-3

10-2

10-1

100

|Vki

ck|(

in k

m-s

-1)

Newtonian1PN1.5PN2PN2.5PN

γf=0.12; γi=0.05

FIG. 2 (color online). Recoil velocity as a function of the parameter � is shown. For all the plots, the value of the parameter �f hasbeen fixed to 0.12 (which corresponds to the final separation of 8:33 Gm=c2 between the two objects under the radial infall). Plots indifferent panels also compare the results with different PN accuracy for four different situations: �i ¼ 0:01, 0.02, 0.05, and 0.0, whichcorrespond to the initial separation (of the two objects in the problem) of 100 Gm=c2, 50 Gm=c2, 20 Gm=c2, and 1 (infinite initialseparation case), respectively.

ERRATA PHYSICAL REVIEW D 87, 069907(E) (2013)

069907-2