SPIN-DOWN MEASUREMENT OF PSR J1852+0040 IN KESTEVEN 79: CENTRAL COMPACT OBJECTS AS ANTI-MAGNETARS

For each observation in Table 2, we transformed the photon arrival times to Barycentric dynamical time (TDB) using the pulsar coordinates determined in Paper II and reproduced in Table 3. Diffuse SNR emission is a significant source of background. To maximize the pulsar signal-to-noise ratio, we used a source extraction radius of 12'' for the XMM-Newton observations, and five columns (24) for the Chandra CC-mode data. An energy cut of 1–5 keV was found to maximize pulsed power. The pulse profile and value of the period in each observation was derived from a Z²₃ periodogram (Buccheri et al. 1983), a choice of harmonics that was found to minimize the uncertainties. The resulting profiles were cross-correlated, shifted, and summed to create a master pulse profile template. Individual profiles were then cross-correlated with the template to determine the time of arrival (TOA) at each epoch.

Table 3. Spin Parameters of PSR J1852+0040

Parameter	Value
R.A. (J2000)a	18h52m3857
Decl. (J2000)a	+00°40'198
Epoch (MJD TDB)b	54597.00000046
Spin period, P	0.104912611147(4) s
Period derivative, $\dot{P}$	(8.68 ± 0.09) × 10−18
Valid range of dates (MJD)	53296–55041
Surface dipole magnetic field, Bs	3.1 × 1010 G
Spin-down luminosity, $\dot{E}$	3.0 × 1032 erg s−1
Characteristic age, τc	192 Myr

Notes. ^aMeasured from Chandra ACIS-I ObsID 1982 (Paper I). Typical Chandra ACIS coordinate uncertainty is 06. ^bEpoch of ephemeris corresponds to phase zero in Figure 2.

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Starting with the dense series of XMM-Newton observations in 2008 September, the TOAs were iteratively fitted using the TEMPO¹ software. We fitted the three observations from September 19–23 to a linear ephemeris and added TOAs one at a time using a quadratic ephemeris, finding that the new TOA would match to <0.1 cycles the predicted phase derived from the previous set. After adding the final observation to the ephemeris, we then worked backward in time from 2008 September to 2004 October until all 23 observations had been included. The quadratic term contributes −8.7 cycles of rotation over the 4.8 year span of the ephemeris, which yields the small uncertainty in $\dot{P}$ discussed below. The phases and errors were determined by cross-correlating with a final iterated template of co-added profiles produced from the best-fitting ephemeris given in Table 3. Figure 1 shows the residuals of the individual observations from the best fit, which demonstrates the validity of the solution. The weighted rms of the phase residuals is 3.4 ms, or 0.032 pulse cycles, which is comparable to the individual measurement errors. There is no evidence of timing noise or higher derivatives in the residuals. The variation of Z²₃ (pulsed power) among the observations listed in Table 2 is also consistent with statistical expectations. Figure 2 shows the summed pulse profile from the 16 XMM-Newton observations, for which it is possible to measure a reasonably accurate background (unlike the Chandra CC-mode data).

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The sensitivity of the results to $\dot{P}$ and B_s can be estimated analytically. If a pulsar is spinning down smoothly, then a coherent timing solution spanning time T will have an uncertainty in frequency of δf = δP/P² ≈ 0.1/T and will be sensitive to a period derivative of $\dot{P}_{\rm \min } = \delta \dot{P} \approx 0.2(P/T)^2$. This in turn will measure $B_s = 3.2 \times 10^{19}\sqrt{P\dot{P}}$ with fractional uncertainty

$$\begin{eqnarray*} {\delta B_s \over B_s} \approx 1.0 \times 10^{38}\ {P^3 \over B_s^2\ T^2} &=& 0.05 \left(B_s \over 10^{10}\,{\rm G}\right)^{-2} \left(T \over 4.8\ {\rm yr}\right)^{-2}\\ &&\times \left(P \over 0.105\ {\rm s}\right)^3. \end{eqnarray*}$$

A coherent ephemeris for PSR J1852+0040 spanning 4.8 year has an uncertainty on $\dot{P}$ of $\delta \dot{P} \approx 1 \times 10^{-19}$, corresponding to a 5σ detection limit of B_s = 5 × 10⁹ G. In comparison, the measured values P = 0.104912611147(4) s and $\dot{P} = 8.68(9) \times 10^{-18}$ have fitted uncertainties consistent with these analytic estimates, and imply in the dipole spindown formalism a surface magnetic field strength B_s = 3.1 × 10¹⁰ G, a spin-down luminosity $\dot{E} = -I\Omega \dot{\Omega }= 4\pi ^2I\dot{P}/P^3 = 3.0\,\,{\times}\,\,10^{32}$ erg s⁻¹, and characteristic age $\tau _c \equiv P/2\dot{P} = 192$ Myr, all with negligible statistical error.

In Papers I and II, we presented several observations of PSR J1852+0040 that were consistent with steady flux and a blackbody spectrum of kT ≈ 0.46 keV. The luminosity of 3.4 × 10³³ d²_7.1 erg s⁻¹ corresponded to a blackbody radius of only ≈0.8 d_7.1 km. This is typical result for a CCO, and indicates a concentrated hot spot that remains to be understood. Using the methods described in Paper I, we now combine all 16 XMM-Newton observations of PSR J1852+0040 into one spectrum in order to search for deviations from a blackbody that might indicate temperature variations on the surface, or cyclotron lines. EPIC pn and MOS spectra were fitted jointly. The accumulated exposure allows us to fit the spectrum over 0.7–7 keV, an increase in coverage over the 1–5 keV range used in the previous papers. While a single blackbody can still be fitted with a temperature of 0.46 keV, the high signal-to-noise ratio and expanded energy range of the summed spectrum reveal systematic deviations from the fit; the reduced χ²_ν = 1.38 for 240 degrees of freedom is unacceptable (see Figure 3 and Table 4). Similar to the results from other CCOs, we find that a fit to two blackbodies is significantly improved (χ²_ν = 1.07), with temperatures of kT₁ = 0.30 keV and kT₂ = 0.52 keV, while the corresponding radii R₁ = 1.9 d_7.1 km and R₂ = 0.45 d_7.1 km still cover only a small fraction of the surface. Here we define radius using the phase-averaged luminosity L_1,2 = 4πR²_1,2σT⁴_1,2 regardless of the unknown emission geometry, which could be, for example, a concentric annulus. Adoption of the two-blackbody fit results in a higher bolometric luminosity than the single blackbody, 5.3 × 10³³ d²_7.1 erg s⁻¹ versus 3.0 × 10³³ d²_7.1 erg s⁻¹. We do not combine the Chandra spectra here for an independent fit because of the increased background and other uncertainties involved in analyzing CC-mode data for this faint source.

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Additional spectral models that were explored include the nonmagnetic neutron star hydrogen atmosphere of Zavlin et al. (1996), and a simplified Comptonized blackbody, as described by Halpern et al. (2008) for application to anomalous X-ray pulsars. Each of these models fits nearly as well as the two-blackbody model because they can eliminate the residual excess at high energy that is left by a single blackbody fit (see Figure 4 and Table 4). For the neutron star atmosphere (NSA), we fixed the parameters M = 1.4 M_☉ and R^∞ = 13.06 km (corresponding to R = 10 km), treating the effective temperature and distance as free parameters. As expected, the fitted temperature is smaller than those of the blackbody models, but the fitted distance of 23.7 kpc is a factor of 3.3 higher than the known value. This indicates that in the atmosphere model only ∼10% of the surface is emitting. We are not motivated to search for an atmosphere model that fits to the full surface area, as was done for the Cas A CCO by Ho & Heinke (2009), because the larged pulsed fraction of PSR J1852+0040 clearly requires a small hot spot. The same interpretation attaches to the Comptonized blackbody model, in which the inferred blackbody radius is only 0.93 d_7.1 km.

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The large intrinsic pulsed fraction, f_p = 64% ± 2% (defined as the fraction of counts above the minimum in the light-curve, corrected for background; see Figure 2) confirms that the emitting area must be small and far from the rotation axis, while the line of sight also makes a large angle to the rotation axis. The pulse shape does not appear to be a function of energy (Figure 5), which also suggests that the emitting area is small. The fitted column density from any model in Table 4 agrees reasonably well with the N_H derived from fits to the SNR spectrum. Sun et al. (2004) find N_H = (1.54–1.78) × 10²² cm⁻² from fitting a variety of equilibrium and non-equilibrium ionization models to ASCA and Chandra spectra of the SNR, while Giacani et al. (2009) find N_H = (1.52 ±  0.02) × 10²² cm⁻² from fitting a non-equilibrium ionization model to the XMM-Newton observations of 2004.

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We then searched for any indication of variability by extracting and fitting the spectrum of each individual observation, fixing the spectral parameters (except for the total flux) to those of the summed spectrum. In order to minimize statistical and systematic errors in this comparison, we restricted the fitted energy range to 1–5 keV for the individual spectra. The resulting individual fluxes are shown in Figure 6, where they are seen to be constant within errors. The mean 1–5 keV flux is 1.9 × 10⁻¹³ erg cm⁻² s⁻¹. The typical 1σ uncertainty in each observation is ≈10%, and the rms dispersion among the 23 observations is 10%.

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Unlike the unique case of 1E 1207.4−5209, there is no evidence for cyclotron absorption lines in the spectrum of PSR J1852+0040 or in any other CCO. 1E 1207.4−5209 has strong absorption features at 0.7 and 1.4 keV (Sanwal et al. 2002; Mereghetti et al. 2002a) that can be attributed to the cyclotron fundamental energy E_c and its first harmonic in a field of ≈8 × 10¹⁰ G (Bignami et al. 2003), according to the relation E_c = 1.16(B/10¹¹ G)/(1 + z) keV, where z ≈ 0.3 is the gravitational redshift. The spectrum of PSR J1852+0040 can now be understood in terms of its weaker surface B-field. For B_s = 3.1 × 10¹⁰ G, the cyclotron fundamental and first harmonic are at 0.27 keV and 0.54 keV, below the region accessible to X-ray spectroscopy (Figure 3), especially because of the larger intervening column density to PSR J1852+0040. The same explanation applied to other well-observed CCOs suggests that they also have weaker surface B-fields than 1E 1207.4−5209.

The steady spin-down of PSR J1852+0040 is consistent with magnetic braking of an isolated neutron star with a weak magnetic field of B_s = 3.1 × 10¹⁰ G. In this case, the derived characteristic age of 192 Myr is not meaningful because the pulsar was born spinning at its current period, P = 0.105 s, as was the case for the other CCO pulsars, 1E 1207.4−5209 and PSR J0821−4300, which have P = 0.424 s and P = 0.112 s, respectively. A recent population analysis of radio pulsars by Faucher-Giguère & Kaspi (2006) favors such a wide distribution of birth periods (Gaussian centered on ∼300 ms, σ ∼ 150 ms).

A B-field of this magnitude could be just the fossil field left by flux conservation during the contraction of the core of the progenitor star, as originally hypothesized for neutron stars by Woltjer (1964). Under this hypothesis, CCOs could simply be those neutron stars in which no additional mechanism of B-field amplification has been effective. As stronger fields can be generated by a turbulent dynamo whose strength depends on the rotation rate of the proto-neutron star (Thompson & Duncan 1993), it is natural that pulsars born spinning slowly would have the weaker B-fields. The model of Bonanno et al. (2006) supports this, in particular finding that the B-field of a slowly rotating neutron star should be confined to small-scale regions, while a global dipole that would be responsible for spin-down is absent.

PSR J1852+0040 falls in a region of ($P,\dot{P}$) parameter space (Figure 7) that is devoid of ordinary (non-recycled) radio pulsars (Manchester et al. 2005)². It overlaps with recycled pulsars in this area. The spin parameters of PSR J1852+0040 are not beyond the empirical or theoretical radio pulsar death lines of Faucher-Giguère & Kaspi (2006) or Chen & Ruderman (1993), respectively. One possible explanation for the absence of radio emission from CCOs is low-level accretion of SN debris for thousands or even millions of years. However, the sample of CCOs is not yet large enough to know if they are intrinsically radio quiet.

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Binary pulsars with similar spin parameters as PSR J1852+0040 are thought to have been partly spun up by accretion. It was also suggested by Deshpande et al. (1995) that most single pulsars in this region are recycled, the birth rate of pulsars with B_s < 10^11.5 G being too low in their estimation, perhaps one in ∼5000 year. Hartman et al. (1997) concluded, to the contrary, that it is not possible to distinguish ordinary pulsars of B_s = 10^10.5−11.5 from recycled ones, and that there is no reason to suppose that any such single pulsar is recycled. The measurements of spin parameters of CCOs certainly increase the inferred birth rate of neutron stars in this weak B-field regime, a renewed warning not to assume that isolated radio pulsars with similar spin parameters are recycled. It should be expected that many of the ≈52 radio pulsars with B_s < 10¹¹ G and 0.1 s < P < 0.7 s in Figure 7 have characteristic ages that are meaningless, being hundreds of millions of years. They have not moved in the ($P,\dot{P}$) diagram and may be former CCOs whose SNRs dissipated only 10⁵–10⁶ yr ago. Once the SNR has disappeared and the neutron star has cooled, it is difficult to recognize and classify such an "orphan CCO" using X-rays unless it was first selected as a radio pulsar. Allowing for this observational handicap, CCOs having weak B-fields may represent a channel of neutron star birth that is as common as any other.

This outcome is counter in its details to the long debated "injection" hypothesis. Using the "pulsar current" analysis proposed by Phinney & Blandford (1981), Vivekanand & Narayan (1981) and Narayan (1987) suggested that a large fraction of pulsars are injected with 0.5 s < P < 1.0 s and $\dot{P} > 1 \times 10^{-14}$ (i.e., with large magnetic field), or that neutron stars do not turn on as radio pulsars until their $\dot{P}$ decreases below a critical value of ∼3 × 10⁻¹³. This would require that they also cool rapidly in order not to be detected as thermal X-ray sources in SNRs. Later investigators (e.g., Lyne et al. 1985; Lorimer et al. 1993) argued that this interpretation suffers from substantial statistical uncertainties and selection effects in pulsar surveys, and that injection of a distinct population of pulsars is not necessary. While previous authors dismissed the possibility of injection of radio pulsars with weak B-fields (e.g., Srinivasan et al. 1984), this appears to be exactly the origin of radio-quiet CCOs. The related problem of the missing pulsars in empty-shell SNRs (Gotthelf 1998) has largely been solved in recent years through a variety of channels, as summarized by Kaspi & Helfand (2002), including the detection of radio-faint pulsars, pulsar wind nebulae in X-rays, and now, pulsations from CCOs.

The spin-down power $\dot{E} = 3.0 \times 10^{32}$ erg s⁻¹ of PSR J1852+0040 is an order of magnitude smaller than its observed thermal X-ray luminosity, L_x ≈ 5.3 × 10³³ d²_7.1 erg s⁻¹, which argues that the latter is mostly residual cooling. Remaining questions presented by the X-ray properties of PSR J1852+0040 and other CCOs are focussed on the details of their X-ray spectra and pulse profiles, which require small heated areas. In the case of PSR J1852+0040, the spectrum is more complex than a single blackbody. In the two blackbody model, for example, the hotter component, of temperature kT₂ = 0.52 keV, has an area of only ∼2.5 d²_7.1 km². The canonical open-field-line polar cap has area A_pc = 2π²R³/Pc ≈ 1.1 km², which is comparable to the area of the hot spot, but polar cap heating by any magnetospheric accelerator must be negligible in this case, being much smaller than the spin-down power. Accretion may cover a wider area and generate more luminosity, but that process is also problematic for PSR J1852+0040, as discussed below. In the remainder of this section, we discuss alternative hypotheses for the weak surface B-fields and thermal hot spots of CCOs, involving internal magnetic fields, anisotropic conduction, and accretion of fall-back material. Although these theories have some attractive properties, they are speculative, and none yet offers a self-consistent explanation for all of the observed properties of CCOs.

While the X-ray luminosity of PSR J1852+0040, 5 × 10³³ erg s⁻¹, is consistent with minimum neutron star cooling scenarios (Page et al. 2004, 2007) for the age of Kes 79 (5.4–7.5 kyr; Sun et al. 2004), the high temperatures and small surface areas fitted to the X-ray spectrum are difficult to understand without invoking either localized heating on the surface or strongly anisotropic conduction. We recall that the X-ray spectra and luminosities of CCOs are not very different from those of quiescent magnetars, another recently recognized class, being attributable largely to one or two surface thermal emission components (e.g., Halpern & Gotthelf 2005). It is difficult to distinguish CCOs from quiescent magnetars without timing data or evidence about variability (Halpern & Gotthelf 2009). Thus, it is tempting to hypothesize that the same magnetic field decay that is thought to be responsible for localized crustal heating in a magnetar can be operating in CCOs. However, in the case of CCOs, such B-fields would need to have the same 10¹⁴–10¹⁵ G strengths as in magnetars to account for persistent X-ray luminosities of 10³³–10³⁴ erg s⁻¹, while having insignificant dipole moments that do not contribute to spin-down. One such configuration would be a small "sunspot" dipole on the rotational equator to account for the large pulse modulation, while the rest of the star contributes dipole B_s < 3 × 10¹⁰ G in order not to exceed the observed spin-down rate. While we cannot rule out this possibility, it would be remarkable if the magnetic field of a neutron star could be created or evolve into a configuration with such extreme contrast. Also, the conspicuous lack of X-ray variability from CCOs argues against invoking the same magnetic field strength and heating mechanism that is held responsible for magnetars, with their ubiquitous variability.

We next turn to the possibility that magnetic field confined entirely beneath the surface can be responsible for the nonuniformity of surface temperature. While a strong B-field is a favored ingredient of models for anisotropic conduction, PSR J1852+0040 with its exceptionally weak dipole field would seem the least likely candidate for such an explanation. Nevertheless, it is possible that a strong toroidal field can exist under the surface, while only a weak external poloidal field contributes to spin-down. A toroidal field is expected to be the initial configuration generated by differential rotation in the proto-neutron star dynamo (Thompson & Duncan 1993). The effects on the surface temperature distribution of an internal toroidal field were calculated by Pérez-Azorín et al. (2006), Geppert et al. (2006), and Page et al. (2007). One of the effects of crustal toroidal field is to insulate the magnetic equator from heat conduction, resulting in warm spots at the poles. It was even shown that the warm regions can be of different sizes due to the antisymmetry of the poloidal component of the field, which is reminiscent of the asymmetric opposing thermal regions in the Puppis A CCO (Gotthelf & Halpern 2009).

In order to explain the large observed pulsed fraction of PSR J1852+0040, the axis of toroidal magnetic field must make a large angle with respect to the rotation axis of the neutron star. An orthogonal configuration was in fact adopted by Geppert et al. (2006) and Page et al. (2007). Even if the toroidal field component is initially parallel to the rotation axis, a toroidal field will deform the neutron star into a prolate shape, which tends toward orthogonality to the rotation axis as a minimum energy configuration (Braithwaite 2009). The main problems with this mechanism are (1) anisotropic conduction is unlikely to produce a small hot region on PSR J1852+0040 that covers only ∼1% of the surface area of the neutron star, and (2) it does not explain why only one of the orthogonal poles is evidently hot.

Furthermore, to have a significant effect on the heat transport, the crustal toroidal field strength required in these models is ∼10¹⁵ G, many orders of magnitude greater than the poloidal field if the latter is measured by the spin-down. Purely toroidal or poloidal fields are thought to be unstable (Tayler 1973; Flowers & Ruderman 1977), although the toroidal field may be stabilized by a poloidal field that is several orders of magnitude smaller (Braithwaite 2009), so this may be a viable configuration for a CCO. On the other hand, twisting and breaking of the crust by such a large toroidal field is the basis of the magnetar model for soft gamma-ray repeaters and anomalous X-ray pulsars (Thompson et al. 2002), which also have large external dipole fields as measured by their rapid spin-down. In this picture, a CCO is a magnetar-in-waiting, a scenario that Pavlov & Luna (2009) found somewhat contrived, as do we. If any pulsar could be an incipient magnetar, it is not explained why CCOs have especially weak surface fields compared to ordinary pulsars.

For as long as the SN explosion mechanism has been studied, it has been noted that a newly born neutron star may accrete large amounts of fall-back material in the hours to months after the SN. This could submerge the initial magnetic field into the core (Geppert et al. 1999), and the surface field could be essentially zero if the accreted matter is not highly magnetized in a well-ordered fashion. After the accretion stops, the submerged field will diffuse back to the surface (Muslimov & Page 1995), but this could take hundreds to millions of years (Geppert et al. 1999). In this picture, the CCOs could be those neutron stars that suffered the most fall-back accretion, while the normal pulsars with ages of a few thousand years must have accreted ≪0.01 M_☉. In the model of Muslimov & Page (1995) with accretion of ∼10⁻⁵ M_☉, the regrowth of the surface field is largely complete after ∼10³ year, so this would probably not explain the weak field of PSR J1852+0040. But if >0.01 M_☉ is accreted, then the diffusion time could be millions of years. Chevalier (1989) calculated that the neutron star in SN 1987A could have accreted ∼0.1 M_☉ of fallback material in the hours after the SN explosion, aided by a reverse shock from the helium layer of the progenitor. If so, it may never emerge as a radio pulsar. A more normal type II SN should accrete 10⁻³ M_☉ or less, which may delay the emergence of the magnetic field for tens of thousands of years, a timescale applicable to explaining the properties of CCOs.

Finally, it should be considered that a thermoelectric instability mechanism driven by the strong temperature gradient in the outer crust (Blandford et al. 1983) could regenerate the magnetic field submerged by accretion. However, the growth of the field via this mechanism could take ∼10⁵ year. CCOs could then be those neutron stars in which the thermoelectric instability is the dominant mechanism of generating magnetic field, perhaps because their initial rotation rates are slow.

The models discussed here have the effect of delaying the emergence of a normal magnetic field for times ranging from a thousand years to essentially forever. They are difficult to test using CCOs, for which it is impossible to measure the braking index, which could indicate field amplification. Furthermore, they do not help to explain the small, hot surface areas seen in the X-ray spectra of CCOs. In order to achieve such a configuration, the magnetic field would have to be submerged everywhere except for one or two small regions, perhaps the magnetic poles of the neutron star, which could remain hot via conduction from the interior.

We also consider ongoing accretion from a fall-back disk of supernova debris as a possible source of anisotropic surface heating. While variability is an indicator of accretion, we do not have evidence of any variations of PSR J1852+0040 at the 10% level among 23 observations spanning 4.8 year, which would tend to eliminate accretion as a significant contributor to its X-ray luminosity. However, although accretion is widely considered to be an inherently unstable process, it is not clear that variability of X-ray binaries or active galctic nuclei can be extrapolated to accretion from a fossil disk at rates of only ∼10⁻⁵ L_Edd. Therefore, we also explore the possibility of accretion, constrained only by the steady spin-down rate of PSR J1852+0040, using the theory of propeller and accretion-disk torques.

The spin parameters of PSR J1852+0040 fall in a regime in which both dipole braking and accretion torques are conceivably significant. In Paper II, we discussed the implications for $\dot{P}$ of PSR J1852+0040 accreting and spinning down in the propeller regime, deriving

$$\begin{eqnarray*} \dot{P} &\approx & 2.2 \times 10^{-16}\,\mu _{28}^{8/7}\,\dot{M}_{13}^{3/7} \left({M \over M_{\odot }}\right)^{-2/7}I_{45}^{-1} \left({P \over 0.105\ {\rm s}}\right)\\ &&\times \left(1- {P \over P_{\rm eq}}\right) \end{eqnarray*}$$

for the propeller effect. In this model, $\dot{M}$ is the rate of mass expelled, which must be ${>} \dot{m}$, which is the accretion rate onto the neutron star. We can therefore assume $L_X = \eta \,\dot{m}\,c^2$ to calculate a lower limit on $\dot{M}$ in the propeller accretion scenario. Assuming efficiency η ∼ 0.1, $\dot{M} > 5 \times 10^{13}$ g s⁻¹ is required. But if we were to adopt μ = B_s R³, where B_s = 3.1 × 10¹⁰ G from assuming dipole spin-down, then the observed $\dot{P} = 8.68 \,\,{\times}\,\, 10^{-18}$ allows a negligible $\dot{M} < 3 \times 10^{8}$ g s⁻¹, which contradicts the accretion assumption.

If instead we use the measured $\dot{P}$ as an upper limit on the propeller spin-down rate, we can derive an upper limit on the required μ in the accretion scenario. Under these assumptions, μ < 3.4 × 10²⁶ G cm³, or B_s < 3.4 × 10⁹ G. However, this would marginally violate the conditions of the propeller model, because for such a small magnetic field, the magnetospheric radius is

$$\begin{equation*} r_m\ =\ 2.3 \times 10^7\,\mu _{26}^{4/7}\,\dot{M}^{-2/7}_{13} \,\left({M \over M_{\odot }}\right)^{-1/7}=\ 2.6 \times 10^7\ {\rm cm}, \end{equation*}$$

which is smaller than the corotation radius

$$\begin{equation*} r_{\rm co}\ =\ ({\it GM})^{1/3}\left({P \over 2\pi }\right)^{2/3} =\ 3.7 \times 10^7\ {\rm cm}, \end{equation*}$$

and the pulsar would be rotating near its equilibrium period. But there has not been enough time in the life of Kes 79 for a weakly accreting pulsar to come into equilibrium. To find PSR J1852+0040 in an equilibrium state would be a remarkable coincidence because its spin-down timescale, $P/\dot{P} \sim 4 \,\,{\times}\,\, 10^8$ yr, is orders of magnitude greater than its age. Therefore, it is more natural to conclude that the small $\dot{P}$ of PSR J1852+0040 is due purely to dipole spin-down, and that it is not accreting. The absence of accretion would also argue against nonuniform surface composition as a cause of the temperature variations. Finally then, none of the possible mechanisms discussed here for hot spots on CCOs is entirely natural and self-consistent.