The Fundamental Plane of Gamma-ray Globular Clusters
C. Y. Hui, K. S. Cheng, Y. Wang, P. H. T. Tam, A. K. H. Kong, D. O. Chernyshov, V. A. Dogiel
aa r X i v : . [ a s t r o - ph . H E ] J a n The Fundamental Plane of Gamma-ray Globular Clusters
C. Y. Hui , K. S. Cheng , Y. Wang , P. H. T. Tam , A. K. H. Kong , D. O. Chernyshov and V. A. Dogiel ABSTRACT
We have investigated the properties of a group of γ -ray emitting globularclusters (GCs) which have recently been uncovered in our Galaxy. By correlatingthe observed γ -ray luminosities L γ with various cluster properties, we probe theorigin of the high energy photons from these GCs. We report L γ is positivelycorrelated with the encounter rate Γ c and the metalicity [Fe / H] which placean intimate link between the gamma-ray emission and the millisecond pulsarpopulation. We also find a tendency that L γ increase with the energy densitiesof the soft photon at the cluster location. Furthermore, the two-dimensionalregression analysis suggests that L γ , soft photon densities, and Γ c /[Fe / H] possiblyspan fundamental planes which potentially provide better predictions for the γ -ray properties of GCs. Subject headings: gamma rays: stars — globular clusters: general — pulsars:general
1. INTRODUCTION
Millisecond pulsars (MSPs) are generally believed as the descenders of the low-massX-ray binaries (LMXBs) (Alpar et al. 1982). As the formation rate per unit mass of LMXBsis orders of magnitude greater in globular clusters (GCs) than in the Galactic field (Katz Department of Astronomy and Space Science, Chungnam National University, Daejeon, South Korea Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong Institute of Astronomy and Department of Physics, National Tsing Hua University, Hsinchu, Taiwan Moscow Institute of Physics and Technology, Institutskii lane, 141700 Moscow Region, Dolgoprudnii,Russia. I.E.Tamm Theoretical Physics Division of P.N.Lebedev Institute, Leninskii pr, 53, 119991 Moscow,Russia (cf.Manchester et al. 2005). The relatively high formation rate of LMXBs and MSPs is a naturalconsequence of the frequent stellar encounters. With the X-ray populations in various GCshave been revealed by the Chandra X-Ray Observatory , Pooley et al. (2003) and Gendre etal. (2003) have found a positive correlation between the number of LMXBs in GCs and thestellar encounter rate, Γ c . This provides evidence for the dynamical formation of LMXBsin GCs. As the descenders of the LMXBs, MSPs are also expected to have a dynamicallyorigin.Very recently, with the corrections of the observational effects in the radio pulsar surveystoward different GCs, Hui, Cheng & Taam (2010) have found a positive correlation betweenthe MSP populations in GCs and Γ c , which has long been predicted. Moreover, the authorshave also found another positive correlation between the metalicity and the MSP population.This relation is not unexpected as the high metalicity in a GC can result in a more efficientorbital shrinkage by magnetic braking. Therefore, the parameter space for the successfulRoche-lobe overflow is enhanced (Ivanova 2006) and subsequently lead to a higher formationrate of MSPs.A brand new window for investigating the MSPs in GCs has been open by the launchof the Fermi Gamma-ray Space Telescope . Since MSPs are the only steady γ -ray emittersin GCs, observations with Fermi can provide an alternative channel for investigating MSPpopulations. Shortly after the commence of its operation, the Large Area Telescope (LAT)onboard the spacecraft has detected γ − rays from 47 Tucanae (hereafter 47 Tuc) (Abdo et al.2009). Terzan 5, which hosts the largest known MSP population, has also been subsequentlydetected (Kong et al. 2010). As the sensitivity of LAT increases monotonically with thecontinuous all-sky survey, a total of 15 confirmed detections of γ − ray emitting GCs havevery recently been reported by Abdo et al. (2010a; 2010b) and Tam et al. (2010). Using ∼ . γ -ray GCs besides 47 Tucand Terzan 5. On the other hand, Tam et al. (2010) have recently reported 7 other newdetections with ∼ γ -ray GCs, Liller 1, which has the highestmetalicity in our Milky way, is also found to have the highest γ -ray luminosity (Tam et al.2010). This discovery further suggests that the effects of metalicity cannot be neglected.Thanks to these surveys, we are able to study these clusters as a unique class for the firsttime.To explain the γ − rays from GCs, there are two main streams. Venter & de Jager (2008)and Venter et al. (2009) suggest the γ − rays are originated from the curvature radiation of ∼ pfreire/GCpsr.html e − /e + pair production and outflow can occur on allopen-field lines. As a result, the outer-magnetospheric gap is quenched by these pairs. Thisscenario is supported by the fact that the MSPs in 47 Tuc are essentially thermal X-rayemitters (Bogdanov et al. 2006). All these demonstrate the potential difficulties of thepulsar magnetospheric model in explaining the observed γ − rays from GCs, which motivatethe exploration of additional / alternative emission mechanisms.On the other hand, Bednarek & Sitarek (2007) have proposed that the inverse Comptonscattering (ICS) could be a possible mechanism to produce γ − rays from GCs. In theirmodel they predict that GCs could be sources of GeV-TeV photons. Unfortunately theyhave ignored the contribution from the Galactic background photons. Very recently, throughgeneralizing the ICS model by including various soft photon fields, Cheng et al. (2010) havefound that the observed γ − ray spectra of all 8 γ -ray GCs reported by Abdo et al. (2010a)can be well-modeled by the ICS between relativistic electrons/positrons in the pulsar wind ofMSPs in the GCs and the background Galactic soft photons. This provides another possibleexplanation for the origin of the γ − rays.In this paper, we report the results from exploring the γ − ray emission properties bycomparing with various cluster properties, which provides us with insight on the origin ofthe γ − rays from this class of GCs. In §
2, we report the method and the results from thecorrelation and the regression analysis. We subsequently discuss the implication of theseresults in §
2. CORRELATION & REGRESSION ANALYSIS
Abdo et al. (2010a) and Tam et al. (2010) have reported 15 GCs with firm detec-tions. This sample size is somewhat larger than that adopted by Pooley et al. (2003) forinvestigating the relations between the X-ray point source populations in GCs and variouscluster parameters. The sensitivity limit of the current sample is ∼ × − erg cm − s − (0.1-100 GeV). The properties of 15 confirmed γ − ray GCs are summarized in Table 1 andthe entries are explained in the following. 4 –Table 1. Properties of the γ − ray emitting GCs.Cluster Name d a Γ cb [Fe/H] c M V d u opticale u IRe log L γ f kpc eV cm − eV cm − erg s − Adopted from Abdo et al. (2010a)47 Tuc 4.0 44.13 -0.76 -9.17 0.93 0.25 34 . +0 . − . Omega Cen 4.8 4.03 -1.62 -10.07 1.61 0.51 34 . +0 . − . M 62 6.6 47.15 -1.29 -9.09 8.07 0.86 35 . +0 . − . NGC 6388 11.6 101.99 -0.60 -9.74 2.59 0.56 35 . +0 . − . Terzan 5 5.5 118.29 0.00 -6.51 7.02 1.37 35 . +0 . − . NGC 6440 8.5 74.17 -0.34 -8.78 10.79 1.00 35 . +0 . − . M 28 5.1 13.10 -1.45 -7.98 5.47 0.92 34 . +0 . − . NGC 6652 9.0 1.24 -0.96 -6.43 3.65 0.51 34 . +0 . − . Adopted from Tam et al. (2010)Liller 1 9.6 77.98 0.22 -7.63 10.53 1.40 35 . +0 . − . M 80 10.3 31.31 -1.75 -8.23 1.88 0.33 34 . +0 . − . NGC 6441 11.7 88.42 -0.53 -9.64 3.59 0.69 35 . +0 . − . NGC 6624 7.9 14.65 -0.44 -7.49 6.03 0.67 35 . +0 . − . NGC 6541 6.9 20.00 -1.83 -8.34 4.69 0.61 34 . +0 . − . NGC 6752 4.4 10.78 -1.56 -7.94 2.01 0.48 34 . +0 . − . NGC 6139 10.1 13.28 -1.68 -8.36 4.10 0.69 35 . +0 . − . . a Cluster distance adopted from Abdo et al. (2010a) and Tam et al. (2010). b Two-body encounter rate estimated by ρ r c σ − with the value scaled with that in M4 whichhas ρ = 10 . L ⊙ pc − , r c = 0 .
53 pc and σ = 8 . c Metalicity d Absolute visual magnitude e Energy densities of various soft photon fields (see text) f γ − ray luminosities adopted from in Abdo et al. (2010a) and Tam et al. (2010) c , metalicity [Fe / H], absolute visual magnitude M V , as well asGalactic background optical / infrared photon densities at the locations of the GCs u optical / u IR .Γ c is the most obvious parameter related to the binary formation rate and hence thenumber of MSP in a GC. This parameter can be estimated as ρ r σ − where ρ is thecentral luminosity density, r c is the core radius and σ is the velocity dispersion at thecluster center. σ are adopted from Gnedin et al. (2002). For ρ and r c , the values are takenfrom Harris (1996; 2003 version) and modified for the distances adopted for this analysis(cf. Tab. 1). Besides Γ c , Hui et al. (2010) have shown that [Fe / H] are also a key parameterin determining the size of the MSP population in a GC. The values of [Fe / H] are takenfrom Harris (1996). On the other hand, if stellar encounters were not the major channelof the binary formation, one would expect the binary population to be correlated with thecluster mass M GC for a primordial binary origin (Lu et al. 2009; Lan et al. 2010). Pooleyet al. (2003) have estimated the cluster mass by integrating the King’s profiles of the GCs.And therefore, their mass estimates are naturally correlated with Γ c . On the other hand,assuming a constant mass-to-light ratio, M GC can also be estimated from the absolute visualmagnitude M V : M GC = 10 − . M V (cf. Hui et al. 2010; Lu et al. 2009). Different from theestimates adopted by Pooley et al. (2003), the correlations between our mass estimates withΓ c is only confident at the level less than 53% (see Figure 1a). Also, the correlation between M V and [Fe/H] only attains a confidence level . M V are also taken fromHarris (1996) and modified for the adopted distances as presented in Table 1.Apart from the number of MSPs, in the context of ICS model, the γ − ray luminosity ofa GC also depends on the energy density of the soft photon field (see Cheng et al. 2010).There are three components of background photons in the Galaxy which can interact withthe relativistic leptons: they are relic, infrared and optical photons. As the energy densityof the relic photons is uniform and does not vary from cluster to cluster, we ignore it in ouranalysis. We obtain the estimates of Galactic optical and infrared photon density, u optical and u IR with the GALPROP code (Strong & Moskalenko 1998).Without a priori knowledge of the distributions of the tested quantities, we follow Pooleyet al. (2003) to adopt a nonparametric correlation analysis. The computed Spearman rankcorrelation coefficients between L γ and various tested quantities are tabulated in Table 2.We have also computed the linear correlation coefficients (i.e. Pearson’s r ) for an intutitiveaccount for the data scattering, though they are less robust than the Spearman ranks inquantifying the correlations. We have also performed the 1-dimensional regression analysis.The best-fit parameters are also given in Table 2. All the quoted uncertainties are 95% 6 –confidence intervals. The best-fit relations of these quantities with L γ are plotted as solidstraight lines in Figure 1b and Figure 2. We have also plotted the upper and lower 95%confidence bands for a visual comparison for the data scattering in each panel.Among all these parameters, the weakest correlation is found for the log L γ − M V rela-tion which is only significant at ∼
15% confidence level. Therefore, there is no convincingcorrelation between these two quantities (see Figure 1b). On the other hand, the correlationsof L γ with Γ c and [Fe / H] are confident at a level over 99% (see Figure 2). All these findingsare fully consistent with the results from analysing the radio MSP population (Hui et al.2010).While the correlation between L γ and Γ c was reported by Abdo et al. (2010a) with8 GCs, the effect of metalicity was ignored in their work. From the 95% confidence bandsshown in Figure 2, the degree of data scattering of the log L γ − [Fe / H] relations is found tobe the smallest among all the tested single parameters. This can be also reflected by thefact that its corresponding linear correlation attains the confidence levels over 99 . L γ also tends to increase with theirenergy densities. For u optical and u IR , the confidence levels for the correlations are > >
99% respectively.In comparison to the log L γ − log Γ c and log L γ − [Fe / H] relations, the relatively largescattering of the data points in the plots of log L γ − log u optical and log L γ − log u IR can be dueto the fact that the GALPROP code is a simplification of the real situation in the Galaxy,which assumes an axisymmetric distribution of all parameters of the program (cf. Strong &Moskalenko 1998). Although it provides reliable average density of background photons inthe Galaxy, the real photon densities at the locations of the GCs can differ by a factor of afew, in particular for those close to the disk.Despite the scattering, the correlation analysis strongly suggests L γ is likely related tothe soft photon energy density estimates. This inference is consistent with the ICS model-ing the γ − ray spectra of GCs (Cheng et al. 2010) which indicates neither the number ofMSPs nor the soft photon energy density is the sole factor in determining L γ . With thisconsideration, we investigate if L γ , Γ c /[Fe / H], and u optical / u IR span a fundamental planeby a 2-dimensional regression analysis. We have examined the sample with the followingrelations: log L γ = a + a log Γ c + a log u optical (1) 7 –log L γ = a + a log Γ c + a log u IR (2)log L γ = a + a [Fe / H] + a log u optical (3)log L γ = a + a [Fe / H] + a log u IR . (4)The best-fit parameters are tabulated in Table 3. We have shown the edge-on view ofthese best-fit fundamental plane relations in Figure 3. In comparison with Figure 2, thedata scatter in these plots are somewhat reduced which suggest these fundmental planerelations can possibly provide us with better γ − ray luminosity predictors than the singleparameter relations. To better constrain the uncertainties of these parameters, we havefurther computed the 1 σ , 2 σ and 3 σ confidence contours for various parametric spaces whichare shown in Figure 4.
3. DISCUSSION
We have examined the γ − ray emission properties of a group of GCs. By investigatingthe possible correlations between the γ − ray power and a number of cluster properties, weshed light on the origin of the γ − rays from these GCs. First of all, the correlation between L γ and Γ c suggests the high energy radiation are intimately related to the population ofdynamically-formed objects, which are presumably MSPs, confirming Abdo et al. (2010a)who used 8 GCs in their study. Together with the lack of any correlation with M V and hencethe cluster mass, this is fully consistent with the inference suggested by Hui et al. (2010)and consolidates the dynamical formation scenario of MSPs in GCs.Apart from Γ c , we have found that L γ is also positively correlated with [Fe / H]. This iswell-consistent with the tendency deduced from studying the radio MSP population in GCs(Hui et al. 2010) and the fact that the GC possesses the highest [Fe/H] also has the highest L γ (Tam et al. 2010). Ivanova (2006) proposes that the absence of the outer convectivezone in metal-poor main sequence donor stars in the mass range of 0 . M ⊙ - 1 . M ⊙ , incomparison to their metal rich counterparts can be responsible, since the absence of magneticbraking in such stars precludes orbital shrinkage, thereby, significantly reducing the binaryparameter space for the production of bright LMXBs. For a conventional scenario, MSPsare the old pulsars that have passed through the death-line in P − ˙ P diagram which are 8 –Table 2. Correlation and 1-D regression analysis of log L γ versus various clusterproperties.Parameters Spearman rank Prob Sa Pearson’s r Prob Pb m c c c log Γ c . ± .
16 34 . ± . / H] 0.7614 0.9990 0.7912 0.9996 0 . ± .
15 35 . ± . M V -0.0536 0.1505 -0.0767 0.2141 0 . ± .
08 35 . ± . u optical . ± .
27 34 . ± . u IR . ± .
44 35 . ± . a The probability that the Spearman rank correlation coefficient is different from zero. b The probability that the linear correlation coefficient (i.e. Pearson’s r ) is different fromzero. c The best-fits for log L γ = mx + c where x is the corresponding parameters listed incolumn 1. Table 3. Best-fit fundamental plane relations of γ − ray GCs.Parameters Best-fit values a . ± . a . ± . a . ± . a . ± . a . ± . a . ± . a . ± . a . ± . a . ± . a . ± . a . ± . a . ± .
50 9 –subsequently spun-up in the binaries. As the metalicity determines the parameter space forsuccessful Roche-lobe overflow, it is also a key parameter in determining the intrinsic numberof MSPs in a GC (Hui et al. 2010; Ivanova 2006).We note that the link between the LMXBs in extragalactic GCs and the metalicity issomewhat weaker than with the cluster mass (e.g. Sivakoff et al. 2007; Kim et al. 2006;Kundu et al. 2002), which is different from the inference drawn from our investigation ofthe Galactic MSP-hosting or γ − ray selected clusters. However, a direct comparison betweenthese two populations has to be cautious. As the MSPs are long-lived and are produced bythe previous generations of LMXBs, their dynamical properties might be different from thatof the LMXB population currently observed. Since the relaxation time at the cluster core isgenerally longer than the lifetime of LMXBs, the cluster is continuously evolved with masssegregation at the cluster center which can result in a varying formation rate of compactbinaries (cf. the discussion in Hui et al. 2010). Also, while a large number of LMXB-hostingGCs in Virgo cluster early-type galaxies have relaxation times > . γ − ray populationwith the best-fits inferred from the radio MSP population in GCs. Hui et al. (2010) havefound that the slopes of log Γ c and [Fe / H] inferred from the radio GC MSPs population are0 . ± .
11 and 0 . ± .
11 respectively. Within these quoted uncertainties, the slope ofthe [Fe / H] relation for the radio population is found to intersect with the 2 σ error contoursfor the corresponding parameters of the γ − ray fundamental plane relations (i.e. a and a ). On the other hand, the logarithmic slope of the Γ c relation for the radio populationis only marginally overlapped with the rims of the 3 σ error contours for the correspondingparameters inferred from the γ − ray population (i.e. a and a ).We have also identified possible positive correlations with various soft photon fieldswhich have significances compatible with those for the encounter rate and the metalicity.These correlations are not expected from the magnetospheric model. Together with theuncertainty of the sustainability of the outergaps in the MSPs in GCs (see § γ -ray from GCs.Abdo et al. (2010a) have argued that the γ -ray emission is magnetospheric in naturebecause of the hard photon indices and the cutoff energies inferred from the phenomenolog-ical model is consistent with the values expected from the magnetospheric model. On the 10 –other hand, Cheng et al. (2010) have recently found that the ICS model can also describe theobserved γ − ray spectra of all the GCs discovered by Abdo et al. (2010a) very well. Simplybased on the model fitting, we were not able to discriminate these two scenarios unambigu-ously. However, different from the case of the magnetospheric model, positive correlationbetween the energy density of the soft photon fields are expected in a ICS scenario as theICS power is directly proportional to soft photon energy density.The energy density of the background soft photon field depends on the location of thecluster. We notice these γ -ray GCs are possibly resided in the Galactic bulge and thereforethey are also metal-rich clusters. This results in a natural correlation between the metalicityand the soft photon energy density with a significance > c , in determining theobserved γ -ray luminosities. This inference is supported by comparing the results reportedby Abdo et al. (2010a) and Tam et al. (2010). Apart from the 8 confirmed cases, Abdoet al. (2010a) have also reported 5 non-detections which include three upper-limits and twoother cases with the γ − ray emission slightly offset from the respective GC cores. With theLAT data of a longer exposure, Tam et al. (2010) have found a larger number of γ -ray GCsincluding 4 previously non-detected cases in Abdo et al. (2010a). This leaves M 15 to bethe only non-detected GC in Abdo et al. (2010a). This is not unexpected from trends ofmetalicity and background soft photon energy density. Although the encounter rate of M 15(Γ c = 53 .
9) is even higher than M 62, its metalicity ([Fe/H]=-2.26) and the background softenergy densities at its location ( u optical = 0 .
44 eV cm − ; u IR = 0 .
11 eV cm − ) are much lowerthan those of all 15 confirmed γ -ray GCs.In view of the aforementioned complication, it is not possible to discriminate the ICSand the magnetospheric scenarios unambiguously simply based on the currently availableinformation. Also, there is still a degeneracy within the context of ICS model. Cheng etal. (2010) show that the γ − ray spectrum from 47 Tuc can be explained equally well byupward scattering of either the relic photons, the Galactic infrared photons or the Galacticoptical photons whereas the γ − ray spectra from the other seven GCs reported by Abdo etal. (2010a) are best fitted by the upward scattering of either the Galactic infrared photonsor the Galactic optical photons. This has prompted us also to discriminate which sourceprovides the predominant soft photon field for ICS.Since the IC radiation power is directly proportional to the energy density of the softphoton field, a logarithmic slope of unity is thus expected for the fundamental plane param-eters a , a , a and a . For the corresponding parameters of u IR (i.e. a and a ), the line of 11 –unity is found to cut through the centers of the 1 σ error contours for both Γ c and [Fe/H] fun-damental plane relations. On the other hand, for the parameters correspond to u optical (i.e. a and a ), the expected value is only marginally intersected with their 3 σ error contours.Although an unambiguous conclusion cannot be drawn from the current population yet, thecomparison between the theoretical expectation and the fundamental plane parameters doesfavor the scenario involving the background infrared emission as the soft photon field for ICupscattering.We would like to point out that the predicted spectral shape in energy regime muchlarger than 10 GeV is significantly different for different soft photon fields (cf. Fig. 2 inCheng et al. 2010). Therefore, observations with TeV facilities, such as MAGIC, HESS andVERITAS, can be feasible to lift up this degeneracy. Furthermore, as the ICS model andthe magnetospheric model predict a rather different TeV spectrum for GCs (Cheng et al.2010; Venter et al. 2009), TeV observations in the future can possibly better discriminatethese two possible contributions of soft photons.Constraints for the emission model can also be derived from the other energy bands.With the diffusion of the relativistic pulsar wind particles, it has been shown that extendedradio and X-ray emission from the GCs can also be produced by synchrotron radiation andICS respectively (Cheng et al. 2010). This is consistent with the recent discovery of thediffuse X-rays around Terzan 5 which exteneded up ∼
10 pc (Eger, Domainko & Clapson2010). Although a clear scenario cannot be identified yet, these diffuse X-rays are more likelyto have a non-thermal origin (Eger, Domainko & Clapson 2010). Assuming these X-rays areoriginated from the tail of ICS, the corresponding γ − ray spectrum can be calculated (Chenget al. 2010). Therefore, a systematic search for the extended X-ray and radio feature outsidethe half-mass radii of the other γ − ray GCs can provide us indpendent constraints.In exploring the fundamental plane relations, our analysis suggests that by combiningthe soft photon energy densities with Γ c /[Fe / H] the data scattering can be reduced. Thesebest-fit relations can provide the indicators in identifying what kind of GCs are potential γ − ray sources for a further search. And the other way round, any deeper γ − ray search fromthe GCs can result in an enlarged sample size and a lower sensitivity limit than the currentvalue (i.e. 6 × − erg cm − s − ), which will certainly enable a further test for all thesereported relations.The authors would like to thank Kinwah Wu and anonymous referee for the usefuldiscussion and providing comments for improving the quality of this manuscript. CYH issupported by research fund of Chungnam National University in 2010. KSC is supported bya GRF grant of Hong Kong Government under HKU700908P. DOC and VAD are supported 12 –by the RFBR grant 08-02-00170-a, the NSC-RFBR Joint Research Project RP09N04 and09-02-92000-HHC-a. And AKHK is supported partly by the National Science Council ofthe Republic of China (Taiwan) through grant NSC96-2112-M007-037-MY3 and a KendaFoundation Golden Jade Fellowship. REFERENCES
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14 –Fig. 1.— a. Two-body encounter rate Γ c vs. absolute visual magnitude M V . b. The γ − rayluminosity L γ vs. M V . The straight line represents the best-fit straight line with the errorsof the data points fully taken into account. The dotted lines represent the upper and thelower 95% confidence bands. 15 –Fig. 2.— L γ vs. various individual cluster properties. The straight lines in the plotsrepresent the best-fits from the linear regression with the errors of the data points fullytaken into account. The dotted lines represent the upper and the lower 95% confidencebands. 16 –Fig. 3.— The edge-on views of the fundamental plane relations of γ − ray GCs. The straightlines in the plots represent the projected best-fits given in Table 3. 17 –Fig. 4.— The χ maps for various parametric spaces of the fundamental plane relations. Thedashed lines illustrate the 1 σ , 2 σ and 3 σ confidence contours for two parameters of interestwhich encircle the best-fit values (i.e. the positions with the lowest χ2