In an attempt to perform a more robust analysis, potentially obtain stronger constraints on the MSP population, and break some parameter degeneracies, we consider a parameterized pulsar spin-down luminosity function 





 (Johnston & Verbunt 1996). We will use this to balance the required X-ray energetics by assuming that 





 and 





. This implies 





, with 





 a normalization constant. Johnston & Verbunt (1996) inferred a typical GC value of γL ∼ 0.5, while Heinke et al. (2006) found γL ∼ 0.4–0.7 for Terzan 5, depending on the energy band. By defining 





, one can next recover the following quantities:
25







26







27







28







29





We want to solve for four quantities: 





, 





, 





 (or, equivalently, 





), and γL; once these are fixed, we can infer the MSP population properties through the above equations. We note, however, that we are using this luminosity function to fit X-ray luminosities, which are integral quantities. We therefore expect to find degenerate solutions, as different combinations might yield the same integral luminosities. Thus, we need four constraints or measurements. We can use the same three constraints as before. Crucially, one needs to specify a fourth parameter ηX to convert from spin-down luminosities to X-ray luminosities. By fixing ηX, we implicitly fix the product 





. As a first attempt, let us assume ηX = 0.05% (e.g., for 





 and 





 erg s−1 to make the calculation consistent with the previous estimate). It is difficult to obtain the actual value for 





, given the effect of the GC cluster potential on the 





 of each MSP (e.g., Bogdanov et al. 2008). If we could, this would further constrain the system via Equation (28). Heinke et al. (2006) noted that while they did not detect an X-ray MSP explicitly, one X-ray source could plausibly be an MSP based on the proximity to a radio MSP position; they also noted that more identifications of X-ray MSPs could be made as radio positions become available. We thus have additional constraints 





 and 





 (which may be used as checks on the consistency of the solutions we obtain). We do obtain a nonunique solution for each fixed value of ηX. However, ηX is not known, and the parameters that satisfy the other three constraints are quite degenerate, as expected. Table 2 indicates a number of parameter combinations that satisfy the observational constraints.25

25
A preliminary Markov chain Monte Carlo investigation (Foreman-Mackey et al. 2013) confirmed the degenerate nature of the free parameters (some are correlated), as well as their being quite unconstrained (reflected by asymmetrical and flat probability distributions, as well as elongated confidence contours). Best-fit values furthermore depend on the choice of priors/parameter bounds. The median values are, however, similar to those in Table 2.
 It is clear that a different choice of ηX will favor a different solution that will imply a different value of 





 (which also depends on the average moment of inertia 





, 





, and 





). For example, a higher value of ηX will yield a lower value of 





 or 





 for a given value of 





 and keeping other parameters fixed. If we require 





 to be the same as assumed in the model used to predict the pulsed emission, this may lead to unrealistic values for γL for a given ηX. Relaxing this requirement (which may easily be done, given other parameter uncertainties) implies more suitable values for the other parameters. It is therefore clear that the system of equations is very coupled and the parameters are degenerate, given the lack of suitable constraints. One may think to constrain the solution space by requiring 





, the latter being the estimated total number of visible MSPs in Terzan 5 as inferred from the Fermi-measured GeV energy flux (Abdo et al. 2010). However, this estimate is quite uncertain and does not contain uncertainties in distance (the square of which determines the γ-ray luminosity Lγ) and conversion efficiency of 





 to Lγ, so this does not seem to be a strong constraint. Likewise, we chose 





 when fitting the CR component, but this value is also subject to other model assumptions, such as MSP geometry and gap width. Finally, it seems that the last few entries in Table 2 might be the more plausible combinations in view of the independent constraints on γL ∼ 0.4–0.7 and 





 (i.e., probably relatively small numbers of visible X-ray pulsars) gleaned from the analysis of Heinke et al. (2006). Thus, the uncertainty in several parameters, particularly ηX, as well as parameter degeneracies preclude us from making definite statements about the MSP population properties. Yet we see that there are several plausible solutions that characterize and constrain the MSP population's energetics, implying that the scenario of MSPs being responsible for the broadband SED may be justified and thus plausible.
Citation: Bogdanov et al. 2008
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