{'fraction_non_alphanumeric': 0.037288904000906724, 'fraction_numerical': 0.03201858778193358, 'mean_word_length': 4.102356512939135, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Luminosities of radio Pulsars

Luminosity is an intrinsic property of radio pulsars related to the properties of the magnetospheric plasma and the beam geometry, and inversely proportional to the observing frequency. In traditional models, luminosity has been considered as a function of the spin parameters of pulsars. On the other hand, parameter independent models like power law and lognormal have been also used to fit the observed luminosities. Some of the older studies on pulsar luminosities neglected observational biases, but all of the recent studies tried to model observational effects as accurately as possible. Luminosities of pulsars in globular clusters and in the Galactic disk have been studied separately. Older studies concluded that these two categories of pulsars have different luminosity distributions, but the most recent study concluded that those are the same. This article reviews all significant works on pulsar luminosities and discusses open questions.


Introduction
Pulsars are rotating magnetized neutron stars which emit beamed electromagnetic radiations from magnetic poles. These objects are amazing tools to test different theories of fundamental physics, including general relativity, alternative theories of gravity, particle physics, nuclear physics, magnetism, plasma physics, etc. Although the emission from these objects ranges from gamma-ray to radio, here I confine myself only to radio pulsars. The first radio pulsar was discovered by Jocelyn Bell in 1967, 1 and since then, there are new discoveries almost every year. A total of 2213 pulsars are listed in the ATNF pulsar catalogue 2 (http://www.atnf.csiro.au/research/pulsar/psrcat/expert.html; March 2013 version), out of which 6 are in the small Magellanic cloud, 15 are in the large Magellanic cloud and 144 are in 28 Galactic globular clusters. Spin periods of these pulsars range from 1.4 ms to 12 s. It is commonly believed that pulsars are born with spin periods of around 1 s and then they slow down by loosing their rotational kinetic energy in the form of electromagnetic energy. Fast pulsars are believed to gain rotational kinetic energy as a result of mass transfer from their binary companion 3, 4 (or past companions, in the cases of isolated fast pulsars). Such pulsars are known as "recycled pulsars" or "millisecond pulsars" (MSPs), and slow pulsars are called "normal" pulsars. Since the exact definition of these categories varies, here I stick to the definition of MSPs as pulsars with spin periods less than 20 ms. Although, pulsars having somewhat higher spin periods (upto around 100 ms) could also gain rotational kinetic energy by accreting matter from their companions; in those cases, I prefer the term "recycled" pulsars. In this review, I try to summarize what has been learnt so far about radio luminosities of pulsars, which is a very important intrinsic property of pulsars and is directly related to the pulsar emission mechanism, the physics of the magnetosphere and the structure of the beam.

Preliminaries on flux densities and luminosities of radio pulsars
Flux density is the amount of energy received from an astronomical object per unit area (of the observing device) per unit time per unit frequency range. The unit of flux density is watts per square metre per hertz. Radio astronomers use the unit jansky (Jy), where 1 jansky = 10 −26 watts per square metre per hertz. As pulsars are comparatively faint objects, their flux densities are usually expressed in milliJansky (mJy). Flux density is a measurable quantity, but it is not an intrinsic property of the object. The intrinsic property of the object is the luminosity, which is the amount of energy radiated by the object per unit time. For an object radiating in a spherically symmetric manner, the luminosity can be written as L = 4πd 2 νup ν low S ν dν, where d is the distance of the object from the observer, S ν is the flux density at any observing frequency ν, ν low and ν up being the lower and upper limits of frequencies over which the object has been observed. But in the case of radio pulsars, the emission is not spherically symmetric; rather, it is beamed from two magnetic poles. So one needs to consider the beaming geometry while defining the luminosity of a radio pulsar as: 5 Here ρ is the radius of the emission cone (assumed circular), δ = W eq /P s is the pulse duty cycle, P s is the spin period of the pulsar, W eq is the equivalent width of the pulse (i.e. the width of a top-hat shaped pulse having the same area and peak flux density as the true profile). As it is usually difficult to determine the values of ρ and δ reliably a , the parameter "pseudoluminosity" is defined as a Moreover, there are different models for the beam structure, like "nested cone", "patchy beam" etc., see Lorimer and Kramer 5 chapter 3.4.3 and references therein for details. In those cases, Eq.
Pulsar Luminosities 3 Note that, although luminosity has the dimension of power, i.e. watts, pseudoluminosity has the dimension of watts per hertz. In pulsar astronomy, pseudoluminosity is expressed in units of mJy kpc 2 , so physical or true luminosity can be expressed in units of mJy kpc 2 MHz. The common practice is to study pseudoluminosity and refer it as luminosity. I follow this convention in the present article (unless otherwise mentioned explicitly), with the subscript implying the value of the observing frequency in MHz, e.g., L 400 stands for the pseudoluminosity of a pulsar observed at 400 MHz and L 1400 stands for the pseudoluminosity of a pulsar observed at 1400 MHz.
It is a well known fact that pulsars are not equally bright at different frequencies. L ν (and as a result S ν ) varies with frequency as L ν ∝ ν α (or S ν ∝ ν α ) and α is known as the "spectral index". The value of α is different for different pulsars. For most of them, it lies in the range of −1 to −2 implying that pulsars are in general brighter at lower frequencies. In Fig. 1, I show the distribution of spectral indices of 294 pulsars listed in the ATNF pulsar catalog (as of March 2013). This distribution has a mean of −1.68 and median −1.70. Extreme examples are PSR J0711+0931 (α = −3.5) and PSR J1740+1000 (α = 0.9). Earlier, Maron et al. 6 found the mean α as −1.8 using 281 pulsars. Toscano et al. 7 obtained a mean α of −1.9 for 19 millisecond pulsars (out of which two are in globular clusters). Bagchi et al. 8 found a mean α of −1.9 for 20 MSPs in globular clusters. They used this value to convert flux densities measured at different frequencies to those at 1400 MHz. They also showed that the luminosity distribution for recycled pulsars in GCs did not vary significantly when α changed in the range of −1.6 to −2.0. This result agreed with the earlier conclusion by Hessels et al. 9 using 37 isolated pulsars in globular clusters. Bates et al. 10 have found that the true distribution of spectral indices for potentially observable radio pulsars can be modelled as a Gaussian with mean -1.4 and standard deviation 1.
3. Luminosity law for radio pulsars: function of the spin period and the rate of change of the spin period The rate of loss of the rotational kinetic energy of a pulsar can be written aṡ whereṖ s is the rate of change of the spin period, Ω = 2π/P s is the angular frequency and I is the moment of inertia of the pulsar.Ė is called the "spin-down luminosity" and it is the total power output by the pulsar. In the simplified assumption that pulsars are magnetic dipoles, rotating in vacuum, this power can be equated to the total electromagnetic power emitted by the pulsar (neglecting energy losses in other forms, e.g., gravitational energy, heat energy, etc.). Even in that case, the total power would be distributed over a wide range of frequency. So it is clear that this spin-down luminosity is a quantity different from the radio luminosity we observe, although according to some theoretical models, thisĖ is related to the physical radio luminosity L (Eq 1) or even with the pseudoluminosity L ν . The luminosity of a radio pulsar at a particular observing frequency ν is commonly expressed by the functional form: where a is a proportionality constant whose dimension depends upon the values of the indices p and q and the units of P s andṖ s . Usually P s is expressed in s andṖ s is s s −1 , but sometimesṖ s is given in units of 10 −15 s s −1 , which changes the value of the constant a. Note that when one tries to fit Eq. (4) to the observed set of pulsars, selection effects play a significant role and the fitted form does not represent the actual population. This form was first used by Gunn and Ostriker 11 who obtained p = 0.905, q = 0.905 for the observed L 400 values of 41 pulsars. This fit can be approximated as p = 1, q = 1. They also proposed an exponential decay of the surface magnetic field b (B S ) with time resulting in a decrease in L 400 with time as B S = 3.2×10 19 P Stollman 17 tried to understand the discrepancy between the luminosity law obtained by different people. He found that p = 1, q = 1 (chosen by Gunn and Ostriker) explained the distribution of pulsars in the B S − P plane better, but p = −1, q = 0.35 (chosen by Prószyński and Przybycień) fitted the observed values of L 400 better. For the first case, he compared a simulated, flux-limited sample of pulsars with the real ones. In his simulation, he chose a normal distribution for the space velocities of pulsars (with a standard deviation of 100 km s −1 ), a lognormal distribution for initial magnetic fields (with a mean of 12.5 and standard deviation of 0.5), a flat distribution between 1-50 ms for initial spin periods, and an exponential decay law for magnetic fields. Soon he proposed a new luminosity law as: 18 Eq (5a) can be approximated to Eq (4) with p = −1.47, q = 0.49 which can be further simplified as p = −1.5, q = 0.5. Stollman 18 also gave a physical explanation for this law within the framework of Ruderman-Sutherland 19 model assuming L 400 to be proportional to the potential difference across the polar magnetospheric gap. All the works mentioned so far in this section fitted observed values L 400 with observed values of P s andṖ s , without considering the sensitivity of pulsar surveys properly, and these fits were used to explain/predict other observable parameters.
Emmering and Chevalier 20 fitted L 400 of observed pulsars by Eq (4) with p = −0.96 ± 0.15, q = 0.39 ± 0.06 which was very close to the fit obtained by Prószyński and Przybycień. They took care of selection effects and performed Mone-Carlo simulations to get the luminosity function for potentially detectable pulsars (through pulsar surveys performed upto that time), as well as the intrinsic pulsar luminosity function. They assumed that the intrinsic luminosity function had a scatter which could be modelled with a normal distribution with the mean fitted with Eq (4), and the measured luminosity was the product of this intrinsic distribution and a lognormal distribution. They obtained different values for p and q depending upon the model, e.g., their model I gave p = −1.11 ± 0.15 and q = 0.43 ± 0.06 for detectable pulsars but p = −1.61, q = 0.64 for the intrinsic distribution. Similarly their model II gave p = −0.82 ± 0.13 and q = 0.31 ± 0.05 for detectable pulsars but p = −1.40, q = 0.43 for the intrinsic distribution. On the other hand, Narayan, 21 who also took care of selection effects, fitted the mean luminosity (L mean 400 ) with p = −1, q = 1/3, a = 10 1.72 (Ṗ s was expressed in 10 −15 s s −1 ) with a scatter of the form 0.2144(1 + cos(1.347X)) where X = log(L 400 /L mean 400 ). Narayan and Ostriker 22 explored different luminosity laws and found that the fit of L mean 400 for normal, isolated pulsars with p = 1, q = 1/3, a = 10 1.635 (Ṗ s in 10 −15 s s −1 ) with a scatter of the form 0.0358 (X + 1.8) 2 exp(−3.6X) gave the best fit. Kulkarni, Narayan and Romani 23 fitted L 400 values of recycled pulsars (including both disk and globular cluster pulsars) with p = 1, q = 1/3, a = 10 1.635 (againṖ s was expressed in 10 −15 s s −1 ) with the same scatter, i.e. 0.0358 (X + 1.8) 2 exp(−3.6X). For the sake of completeness, I quote two other models used by Narayan and Ostriker. 22 The first one was with a scatter as 0.0340 (X + 1.9) 2 exp(−3.3X). Afterwards, with a larger dataset (412 pulsars) and an improved model to estimate the distances of the pulsars, Lorimer et al. 24 found that neither p = 1, q = 1 nor p = −1, q = 1/3 adequately described the observed values of L 400 , their best fit values were p = −0.68 ± 0.12, q = 0.28 ± 0.05. On the other hand, when they used the luminosity laws by Stollman 18 and by Emmering and Chevalier 20 with a Gaussian spread of 0.8 as the intrinsic distribution of in log L 400 , and performed Monte-Carlo simulations, they obtained plausible fits to the P s , B S , L 400 , d and DM distributions of the observed pulsar populations. Here DM is the dispersion measure, i.e. the free electron column density integrated along the line-of-sight, and is expressed in the unit of pc cm − 3 Malov, Malov and Malofeev 25 obtained a fit (without considering any selection effect) as They also obtained a relation of L 400 with B LC /P 2 where B LC is the magnetic field near the light cylinder, as Some of the works mentioned in this section, e.g., Narayan, 21 Emmering and Chevalier, 20 Narayan and Ostriker, 22 Lorimer et al., 24 are early examples of population synthesis which has become very popular in pulsar astronomy nowadays. In the case of a population synthesis, first one needs to use appropriate birth distributions of P s ,Ṗ s (or P s , B S ), which result a distribution of L ν . Then one needs to model other parameters like the space distribution of pulsars at birth, their motion through the Galactic potential, evolution of the spin parameters, etc. Finally through modeling of pulsar surveys (i.e. detection sensitivity, sky coverage etc.), one obtains synthetic distributions of parameters like P s ,Ṗ s , L ν (or S ν ) for "observable" pulsars which are compared with the observed values to test the validity of the models.
So far, I have discussed only models for pseudoluminosity, but there are efforts to model the physical or true luminosity of radio pulsars as defined in Eq. (1). Arzoumanian, Chernoff, and Cordes 26 modelled In the next section, I discuss efforts to understand the distribution function of pseudoluminosities (which I again mention as luminosity) of radio pulsars, independent of spin parameters.

Luminosity function for radio pulsars
The luminosity function of any specific type of astronomical objects is defined as the number of such objects per unit luminosity interval. This can be done either by using a population synthesis method or by directly fitting the observed luminosities. In this section, first I discuss the direct method and then the population synthesis method used by various people to obtain the luminosity function for pulsars in the Galactic disk. Efforts to obtain the luminosity function for the pulsars in globular clusters will be discussed separately.

Direct method
The most popular luminosity function used for radio pulsars is the power law, which can be written as where ρ(L ν ) is the number or density of pulsars with luminosities in the range of L ν to L ν + dL ν and ρ 0 is the constant of proportionality. Commonly, the interval dL ν is taken as the unit logarithmic interval (as the observed values of L ν spread over several orders of magnitudes). Large 28  Moreover, in these studies, ρ(L ν ) was chosen as the space density of pulsars in the luminosity interval L ν to L ν + dL ν , so to obtain the number of pulsars in that interval, integration over the volume was required. But the use of ρ(L ν ) as the number of pulsars in the luminosity interval of L ν to L ν + dL ν is equally valid. ρ(L ν ) can be converted to a probability distribution function (PDF) with proper normalization, i.e. by adjusting the constant ρ 0 and can be written as: where L ν,min is the minimum value of L ν . Equation (12) leads to the cumulative distribution function (CDF) as and the complementary cumulative distribution function (CCDF) as F c pl (L ν ) is the probability that the value of the radio luminosity at the observing frequency ν is greater than L ν . But the number of pulsars having luminosities greater than or equal to L ν should be expressed with the complementary cumulative frequency distribution function (CCFDF) as: where N tot = ∞ Lν,min ρ(L ν ) dL ν is the total number of pulsars and N ( ν,min , and β = γ + 1. It is clear from Eq. (15) that N (≥ L ν ) = N 0 for L ν = 1 mJy kpc 2 . Now I discuss the studies where people fitted Eq. (15) to pulsar luminosities.
Allakhverdiev, Guseinov, and Tagieva 34 fitted the CCFDF of L 400 with β = −0.9 ± 0.1 and N 0 = 562 for 68 pulsars with L 400 > 0.3 mJy kpc 2 . Guseinov et al. 35 tried to find accurate luminosity functions for both L 400 and L 1400 using only pulsars closer than 1.5 kpc to obtain a sample with large enough numbers of low luminosity pulsars. They fitted broken power-laws for both L 400 and L 1400 . For L 400 , they found and for L 1400 , they found It is clear that in the above fits, the low luminosity part was much flatter for L 400 than that for L 1400 . The authors suggested that the incompleteness of the pulsar surveys at 400 MHz was responsible for this result. They also obtained luminosity functions for only isolated pulsars with characteristic ages < 10 7 yr as With a larger dataset (412 pulsars), an improved model to estimate the distances of the pulsars, and a Monte-Carlo method, Lorimer et al. 24 found that the distribution of total galactic population of pulsars with L 400 > 10 mJy kpc 2 could be expressed as N (≥ L 400 ) = (7.34 ± 1.06) × 10 4 L −1 400 and the distribution of potentially observable pulsars with L 400 > 10 mJy kpc 2 could be expressed as N (≥ L 400 ) = (1.31 ± 0.17) × 10 4 L −1 400 . Using the flux densities and distance estimates available in the ATNF catalogue in 2010 for 51 MSPs (P s < 20 ms) in the Galactic disk, Hui, Cheng and Taam 36 fitted the CCFDF with Eq. (15) for different subcategories, i.e. the isolated, binary, and total population. I quote their best fit parameters in Table 1. They assumed N (≥ L 1400 ) as the uncertainties in the data (Poisson noise), and fitted log N (≥ L 1400 ) with a linear regression analysis. The power law became steeper when they kept only pulsars L 1400 ≥ 1.5 mJy kpc 2 , except for the case of isolated pulsars (which had a very small sample size).
I re-perform the analysis with the L 1400 values available in the ATNF catalogue in March 2013 for MSPs (P s < 20 ms) in the Galactic disk, and the best fit parameters are given in Table 2. Note that although the fitting coefficients are different from what obtained by Hui, Cheng and Taam, the fact that the power law become steeper when only pulsars with L 1400 ≥ 1.5 mJy kpc 2 are considered, remains the same. I also fit the distribution of L 1400 keeping mildly recycled pulsars, i.e. taking all the pulsars with P s < 100 ms, and the fit parameters are given in Table 3. For both of the cases, the total population can be fitted better with a double power law, as: The best fit parameters are L break = 2.24 mJy kpc 2 , N 0l = 99 +1

Population synthesis method
Faucher-Giguère and Kaspi 37 performed a detailed population synthesis study of isolated, normal pulsars in the Galactic disk (the basic scheme was the same as      what has been described in the antepenultimate paragraph of Section 3 of this article). They used two different luminosity laws for L 1400 . The first one was a broken power law, independent of pulsar parameters, as follows: with parameters L low 1400 = 0.1 mJy kpc 2 , L to 1400 = 2.0 mJy kpc 2 , γ 1 = −1.267, and γ 2 = −2.0. The second one was the conventional P s −Ṗ s law as: where L 1400,corr was chosen from a zero-centered normal distribution with a standard deviation of σ L1400,corr . The best fit values of the parameters were σ L1400,corr = 0.8, A 0 = 0.18 mJy kpc 2 , p = −1.5, and q = 0.5. By comparing the parameters of simulated (and selected) pulsars with the real ones, they noticed that the broken power law (Eq. 21) did not work well, specifically, it lead to too high values for the scale height (perpendicular distance from the Galactic disk) and produced a large number of pulsars near the "death line" in the P s −Ṗ s diagram. On the other hand, the P s −Ṗ s dependent luminosity law (Eq. 22) worked fine, but interestingly, they noticed that the simulated distribution of L 1400 using this model could be simplified by a base-10 lognormal distribution, for which the PDF can be written as: f lognormal (L 1400 ) = log 10 e L 1400 Here µ is the mean of the distribution of log 10 L 1400 and σ is the standard deviation. The best fit values of these parameters were µ = −1.1 and σ = 0.9, which translated to the mean of the distribution of L 1400 as 10 −1.1 mJy kpc 2 or 0.079 mJy kpc 2 . Ridley and Lorimer (2010) 38 performed almost similar study with much exhaustive models for the spin evolution of the pulsars and arrived at the same conclusion. Afterwards, Bagchi et al. 8 concluded that this lognormal luminosity model worked fine for luminosities of recycled pulsars in the Galactic globular clusters, although they could not constrain the parameter space very well, which was later tried by Chennamangalam et al. (2013). 39 These two works will be discussed in details in the next section.

Luminosity function for pulsars in globular clusters
For pulsars in globular clusters (GCs), it is difficult to model the effects of stellar encounters and the cluster potential. 40 So the population synthesis method to study the luminosity distribution of cluster pulsars is not popular, the direct method is commonly used.
Anderson 41  Similar to their analysis for MSPs in the Galactic disk, Hui, Cheng and Taam 36 fitted a power law (Eq. 15) for GC pulsars. They selected only such GC pulsars, for which each GC had atleast four pulsars with published values of flux densities. They used a spectral index of −1.8 to convert flux densities measured at other frequencies to 1400 MHz and the distances of globular clusters published in Harris catalogue (http://physwww.physics.mcmaster.ca/∼harris/mwgc.dat) 47 to obtain L 1400 . I quote their best fit parameters in Table 4. The power law became steeper when only pulsars with L 1400 ≥ 1.5 mJy kpc 2 were considered. I also compare these fit parameters with the ones obtained by Hessels et al. 9 (only β, values of N 0 were not reported by Hessels et al.).
Hui, Cheng and Taam 36 also concluded that the luminosities of MSPs in GCs were different from those in the Galactic disk as the CCFDF for GC MSPs was much steeper than that of disk pulsars (by comparing Table 1 with Table 4). This is a very important conclusion. If correct, it would imply that the radio luminosity is related to differences in formation processes between the disk and GC pulsars. The same analysis was re-performed by Bagchi and Lorimer 48 with more recent distance estimates of GCs and the resultant CCFDF was even steeper. They obtained N 0 = 59 +1 −1 and β = −0.80 ± 0.03 when they kept all pulsars and N 0 = 74 +5 −4 and β = −1.06±0.06 when they kept only the pulsars with L 1400 ≥ 1.5 mJy kpc 2 . Remember that although for disk pulsars, only pulsars with P s < 20 ms were chosen, no such criterion was chosen for GC pulsars by Hui, Cheng and Taam; and Bagchi and Lorimer used a condition P s < 100 ms. Hui, Cheng and Taam also fitted power laws for pulsars in different GCs separately keeping only pulsars with L 1400 ≥ 0.5 mJy kpc 2 , which was also redone by Bagchi and Lorimer. The comparison is in Table 5. Again, we need to remember the difference between two datasets. Hui, Cheng and Taam did not exclude pulsars with P s ≥ 100 ms, which Bagchi and Lorimer did. Moreover, Bagchi and Lorimer used latest distance estimates for GCs to convert S 1400 to L 1400 . The largest discrepancy in the value of the distance was for the case of  At this time, flux estimates for the pulsars in NGC 6517 have been published and were included in the later study by Bagchi et al. 8 Here, for the sake of comparison, I re-perform the analysis similar to Bagchi and Lorimer 48 with this improved dataset. Interested readers can consult Tables 1 and 4 of Bagchi et al. 8 for details of pulsar parameters (including P s , S 1400 , cluster association etc.) and cluster parameters. The best fit parameters for pulsars with P s < 100 ms are given in Table 6 and the best fit parameters for pulsars with P s < 20 ms (which was the condition used for disk MSPs) are given in Table 7. Table 7 (which shows the best fit parameters for GC MSPs) should be compared with Table 2 (which shows the best fit parameters for disk MSPs). For each category, i.e. the binary, isolated and total population, the luminosity distribution of MSPs in GCs is much steeper than that of the MSPs in the Galactic disk. A double power law (Eq. 20) still gives a better fit than a single power law. The best fit parameters are L break = 1.78 mJy kpc 2 , N 0l = 63 +1 −1 , β l = −0.51 ± 0.05, N 0h = 88 +9 −8 , and β h = −1.16 ± 0.08 if I keep all pulsars (with P s < 100 ms) and L break = 2.82 mJy kpc 2 , N 0l = 71 +9 −8 , β l = −0.93 ± 0.18, N 0h = 117 +31 −25 , and β h = −1.32 ± 0.15 if I keep only pulsars with L 1400 ≥ 1.5 mJy kpc 2 (and P s < 100 ms). Fig. 3 shows the fits. For only short period pulsars (P s < 20 ms), I get L break = 1.6 mJy kpc 2 , N 0l = 59 +1       As we already know, the direct method to obtain the luminosity function does not take care of selection effects, and as GCs are in general at large distances, it is difficult to detect low luminosity pulsars in GCs, thus the observed sample is more biased towards bright pulsars, than that for the disk pulsars. To overcome this limitation, Bagchi et al. 8 introduced an improved method where they modelled the luminosity distribution of recycled pulsars in globular clusters as the brighter, observable part of an intrinsic (parent) distribution, assuming that the parent luminosity distribution is the same for all GCs. They considered the sample of 83 pulsars (with known flux values) with spin periods P ≤ 100 ms in 10 GCs with the condition that each of these GCs hosted at least 4 such pulsars. For all these objects, the spin and binary properties suggested that the neutron stars had undergone the phase of recycling in the past. They converted flux densities measured at other frequencies to S 1400 using the estimated values of α from observed values of fluxes at different frequencies whenever available, otherwise α = −1.9. Details of flux densities and spectral indices for GC pulsars can be found in Table 1 of Bagchi et al. They used latest distance estimates of GCs to obtain L ν from S ν (Eq. 2). It is a well known fact that the measured values ofṖ s for GC pulsars are affected by cluster potentials, 40 and Bagchi et al. 8 did not find any correlation between L 1400 and P s for GC pulsars. That is why instead of using any P s ,Ṗ s dependent luminosity law (as described in Section 3), they used simple luminosity distribution functions. Using those distribution functions, they generated synthetic samples of pulsar luminosities in each GC and selected only those pulsars which had simulated luminisities greater than the minimum observed luminosity for that GC, until they obtained the desired number of such bright and selected pulsars in each case. The number of pulsars they needed to generate to obtain such desired number of bright pulsars gave estimates of total number of pulsars in each GC, and the sum of the fluxes of all these pulsars gave estimates of total fluxes from each GC. They combined luminosities of all such bright pulsars to get a combined luminosity distribution which they compared with the observed distribution. A key assumption in this method was that each GC had been searched down to the level of the faintest observable pulsar in that particular cluster. This assumption provided a good approximation to the actual survey sensitivity in each cluster, and was made primarily due to the lack of published detail of several of the globular cluster surveys.
The first distribution function chosen by Bagchi et al. was the lognormal luminosity function (see Eq. 23 for the PDF). They chose both µ and σ as free parameters and perform their analysis (as mentioned in the above paragraph) for each set of values of µ, σ. To compare the simulated luminosity distributions with the observed one, they performed two statistical tests, the KS test and the χ 2 test. For both of the tests, they obtained wide ranges of values of µ and σ providing good fits. For examples, they quoted that the KS test resulted the best fit (maximum value of P KS = 0.98) for µ = −0.61 and σ = 0.65, while the χ 2 test resulted the best fit (minimum value of χ 2 = 6.3) for µ = −0.52 and σ = 0.62. The 2-σ contour around the minimum χ 2 and the region of P KS ≥ 0.05 were almost the same and enclosed large areas in the µ − σ space. The model used by Faucher-Giguère and Kaspi 37 for isolated, slow pulsars in the Galactic disk (µ = −1.1 and σ = 0.9) fell inside those regions, i.e. provided a good fit. The next distribution they used was the power law c , for which the PDF is given in Eq (12). They chose L 1400,min in the range of 0.003 − 0.48 mJy kpc 2 , as 0.48 mJy kpc 2 was the observed minimum luminosity among GC pulsars in their sample, and the lower value of L 1400,min was chosen somewhat arbitrarily. They also chose a maximum luminosity of 50 mJy kpc 2 and argued that there was no GC pulsar with L 1400 > 20 mJy kpc 2 , and their results were insensitive to the exact choice of the maximum luminosity cutoff over the range of 20-500 mJy kpc 2 . Here the KS test resulted the best fit (maximum value of P KS = 0.81) for γ = −1.92, L 1400,min = 0.017 mJy kpc 2 and the χ 2 test resulted the best fit (minimum value of χ 2 = 8.0) for γ = −2.01, c Remember β = (γ + 1) according to the notation used in this article, while Bagchi et al. used the notation β = −(γ + 1).
L 1400,min = 0.022 mJy kpc 2 . Based on this, they concluded that the lognormal function was statistically slightly better description for the luminosity distribution of recycled pulsars in GCs. They also demonstrated that an exponential distribution with PDF f exponential (L 1400 ) = λe −λL1400 did not work well for any value of the parameter λ (1/λ is the mean of the distribution). They also predicted the total number of pulsars and total flux densities for their best fit models (see Tables 2  and 3  Bagchi et al. noticed that all their best models (mentioned earlier) could reproduce the observed diffuse flux for 47 Tuc. For Terzan 5, the power law models provided better matches to the diffuse flux overall, while the lognormal models predicted slightly smaller fluxes which lied 2-5σ below the nominal value found by Fruchter and Goss. 44 To constrain the parameter space further, first they worked on the lognormal distribution, where they fixed µ as −1.1 and varied σ. They noticed that there were only two possible ranges of σ which were compatible with the diffuse flux from Terzan 5: σ ∼ 0.5 or σ ∼ 0.9, but they preferred the "solution" with σ ∼ 0.9 as the set µ = −1.1, σ = 0.5 did not provide good χ 2 fitting. Remember, µ = −1.1, σ = 0.9 is the parameter set preferred by Faucher-Giguère and Kaspi for isolated, normal pulsars in the Galactic disk. Similarly, for the power law distribution, fixing γ = −2, they found that a wide range of values of L 1400,min were consistent with the diffuse flux from Terzan 5. As the lower end of this range gave an unrealistically large number of pulsars, they preferred the upper end of this range, which corresponded to L 1400,min = 0.05 mJy kpc 2 . Note that this value of L 1400,min is higher than the presently known lowest value of L 1400 = 0.01 mJy kpc 2 for the disk pulsar J1741-2054 (Camilo et al. 49 ). Bagchi et al. did not find any correlation between the predicted number of pulsars in different GCs and GC parameters, but with the improved estimates of stellar encounter rates, Bahramian et al. 50 demonstrated that the number of pulsars as predicted by Bagchi et al. was correlated with the stellar encounter rates.
Chennamangalam et al. 39 employed a Bayesian technique to further constrain the parameters of the lognormal distribution. They treated pulsar populations in different globular clusters separately unlike most of the previous works where people studied the total pulsar (or recycled pulsar) population in all clusters. Moreover, they first performed their study in the flux domain and then translated the results to the luminosity domain. In addition to flux values of individual pulsars, they used the values of total diffused flux also. They studied three clusters, Terzan 5, 47 Tuc and M 28, as these are the top three according to the number of pulsars with known flux estimates, and total diffuse flux values are also known. Note that although they did not explicitly put any condition on the spin period of the pulsars, their dataset contained only recycled (for a few cases though mildly) pulsars. Table  8 summarizes the data and priors they used and the posteriors they obtained. The values of flux of individual pulsars and total diffuse flux values (S 1400,tot ) for each GC were taken from the literature. See Table 1  that the luminosity distribution for the disk pulsars and the cluster pulsars are the same (Ridley and Lorimer studied disk pulsars). For each cluster, the prior in S min was taken as uniform in the range 0 to the minimum of the measured pulsar fluxes for that cluster. Interestingly, osteriors in µ and σ were different for different clusters. Although Chennamangalam et al. succeeded to narrow down the allowed ranges of µ and σ, their allowed ranges for number of pulsars were much larger than those obtained by Bagchi et al. There is scope for improving both these analyses by including more data points (whenever more flux values will be available) and by incorporating uncertainties in flux measurements.

Summary and Conclusion
In this review, I tried to cover all the significant works on luminosities of radio pulsars. The main open question at this point is that if there is no difference between the luminosity distribution for normal isolated pulsars in the Galactic disk and recycled pulsars in GCs (as found by Bagchi et al.), then why the difference in the luminosity distributions of pulsars in different GCs has been observed by Chennamangalam et al.? Insufficient data may have biased any one or both of these studies. The second question is that whether luminosities of radio pulsars depend on the values of P s andṖ s , and if yes then what is the exact form and what is the physical explanation? Why the lognormal function provides a better fit than the conventional power-law? What are the exact parameters for the lognormal distribution? We do not know answers to these questions and do not know how much the past studies have been affected by observation limitations. So continuous study on this topic is needed.