Erratum: “Constraints on Undetected Long-period Binaries in the Known Pulsar Population” (2023, ApJ, 951, 20)

In the published article, we explored the presence of undetected long-period ( where the orbital period is much greater than the data span ) binaries in the existing pulsar catalog. These would be systems where binary motion was instead modeled as a series of secular frequency derivatives. While we did not identify any likely binary systems, we did set out a methodology for evaluating candidates based on their frequency derivatives and found that long-period ( 2 kyr ) binaries could be present in as much as 35% of the population based on observed period derivatives and limits. However



( ) where x is the projected size of the pulsar's semimajor axis, κ and ò are terms proportional to the eccentricity, and is the orbital phase (mean anomaly, to within a constant offset) as a function of time.
Here n b = 2π/P b is the orbital frequency for period P b , and the time of the ascending node can be computed as where T 0 is the time of periastron (also see Lorimer & Kramer 2012).Taking the circular case (ò = κ = 0), we find We define t = T − T 0 and write the circular Römer delay as From this result, we can get the line-of-sight velocity, and so on.We then look at Joshi & Rasio (1997), where their Equation (3) states where c is the speed of light, the spin frequency is f, and so on.To compute the values of the apparent frequency derivatives now, we evaluate Equations (8) and (9) at t = 0 to find Overall, we find the nth frequency derivative to be Note that this differs from Equation (8) in the published article: the n! in the denominator is not present, the signs are flipped, and the dependence on ω is flipped between sin and cos (see Table 1 for an explicit comparison).
The sign swap and the sin/cos swap are not consequential for the conclusions of the published article, since a replacement of ω → − ω − π/2 will accomplish that.This means that ω would be redefined but since it is a nuisance variable, it does not change the conclusions.The ratios between successive terms keep the correct behavior.
The lack of n! in the correct terms does change the conclusions slightly, but not qualitatively.For instance, constraints on f f 2 = would change by a factor of 2, and using the ratio between successive terms to constrain P b (Equation ( 13) of the published article) would be correct except for the removal of n + 1 in the denominator.The majority of the figures in the published article are essentially unchanged.The exception is Figure 4 (a revised Figure 4 is presented here).One of the pulsars (PSR J1913+1330) included in that figure cannot be solved using the correct expressions for the frequency derivatives as the signs of the measured  prefactor given in the published article is written differently but has the same magnitude.
f 2 /f 3 /f 4 are not consistent, but one of the pulsars originally excluded is solvable.
We have verified our calculations a python script to simulate an ELL1 binary and fit it with just frequency derivatives in PINT (Luo et al. 2021).The results are consistent with those presented here.
This is the same as given in (8.26) ofLorimer & Kramer (2012), although here we only give it for zero eccentricity.We can also identify further derivatives:

Figure 4 .
Figure 4. Top: for PSR J1929+1357, instead of allowing ω < 0.52 we find ω > 0.52.Bottom: for PSR J1910+0517 (originally excluded from this figure), we identify a potential solution with ω = 0.39 ± 0.06 rad and P b = 3.4 ± 0.2 yr.The published version of this figure also included PSR J1913+1330, but that does not permit a solution as ⃛ f and ⃜ f should have the same sign.

Table 1
Comparison of Frequency Derivative Terms with Those from the Published Article