Asymptotic enumeration of dense 0-1 matrices with specified line sums

Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of non-negative integers with sum_{i=1}^{m} s_i = sum_{j=1}^n t_j. Let B(S,T) be the number of m*n matrices over {0,1} with j-th row sum equal to s_j for 1 <= j <= m and k-th column sum equal to t_k for 1 <= k <= n. Equivalently, B(S,T) is the number of bipartite graphs with m vertices in one part with degrees given by S, and n vertices in the other part with degrees given by T. Most research on the asymptotics of B(S,T) has focused on the sparse case, where the best result is that of Greenhill, McKay and Wang (2006). In the case of dense matrices, the only precise result is for the case of equal row sums and equal column sums (Canfield and McKay, 2005). This paper extends the analytic methods used by the latter paper to the case where the row and column sums can vary within certain limits. Interestingly, the result can be expressed by the same formula which holds in the sparse case.Comment: Multiple minor adjustments. Accepted by JCT-

Pulse Profiles, Accretion Column Dips and a Flare in GX 1+4 During a Faint State

The Rossi X-ray Timing Explorer (RXTE) spacecraft observed the X-ray pulsar GX 1+4 for a period of 34 hours on July 19/20 1996. The source faded from an intensity of ~20 mCrab to a minimum of <~0.7 mCrab and then partially recovered towards the end of the observation. This extended minimum lasted ~40,000 seconds. Phase folded light curves at a barycentric rotation period of 124.36568 +/- 0.00020 seconds show that near the center of the extended minimum the source stopped pulsing in the traditional sense but retained a weak dip feature at the rotation period. Away from the extended minimum the dips are progressively narrower at higher energies and may be interpreted as obscurations or eclipses of the hot spot by the accretion column. The pulse profile changed from leading-edge bright before the extended minimum to trailing-edge bright after it. Data from the Burst and Transient Source Experiment (BATSE) show that a torque reversal occurred <10 days after our observation. Our data indicate that the observed rotation departs from a constant period with a Pdot/P value of ~-1.5% per year at a 4.5 sigma significance. We infer that we may have serendipitously obtained data, with high sensitivity and temporal resolution about the time of an accretion disk spin reversal. We also observed a rapid flare which had some precursor activity, close to the center of the extended minimum.Comment: 19 pages, 6 figures, accepted for publication in Astrophysical Journal (tentatively scheduled for vol. 529 #1, 20 Jan 2000

Accretion column eclipses in the X-ray pulsars GX 1+4 and RX J0812.4-3114

Sharp dips observed in the pulse profiles of three X-ray pulsars (GX 1+4, RX J0812.4-3114 and A 0535+26) have previously been suggested to arise from partial eclipses of the emission region by the accretion column occurring once each rotation period. We present pulse-phase spectroscopy from Rossi X-ray Timing Explorer satellite observations of GX 1+4 and RX J0812.4-3114 which for the first time confirms this interpretation. The dip phase corresponds to the closest approach of the column axis to the line of sight, and the additional optical depth for photons escaping from the column in this direction gives rise to both the decrease in flux and increase in the fitted optical depth measured at this phase. Analysis of the arrival time of individual dips in GX~1+4 provides the first measurement of azimuthal wandering of a neutron star accretion column. The column longitude varies stochastically with standard deviation 2-6 degrees depending on the source luminosity. Measurements of the phase width of the dip both from mean pulse profiles and individual eclipses demonstrates that the dip width is proportional to the flux. The variation is consistent with that expected if the azimuthal extent of the accretion column depends only upon the Keplerian velocity at the inner disc radius, which varies as a consequence of the accretion rate Mdot.Comment: 7 pages, 5 figures, accepted by MNRAS. Included reference

The supermassive black hole in the Seyfert 2 galaxy NGC 5252

We present results from HST/STIS long-slit spectroscopy of the gas motions in the nuclear region of the Seyfert 2 galaxy NGC 5252. The observed velocity field is consistent with gas in regular rotation with superposed localized patches of disturbed gas. The dynamics of the circumnuclear gas can be accurately reproduced by adding to the stellar mass component a compact dark mass of MBH = 0.95 (-0.45;+1.45) 10E9 M(sun), very likely a supermassive black hole. Contrarily to results obtained in similar studies rotational broadening is sufficient to reproduce also the behaviour of line widths. The MBH estimated for NGC 5252 is in good agreement with the correlation between MBH and bulge mass. The comparison with the MBH vs sigma relationship is less stringent (mostly due to the relatively large error in sigma); NGC 5252 is located above the best fit line by between 0.3 and 1.2 dex, i.e. 1 - 4 times the dispersion of the correlation. Both the galaxy's and MBH of NGC 5252 are substantially larger than those usually estimated for Seyfert galaxies but, on the other hand, they are typical of radio-quiet quasars. Combining the determined MBH with the hard X-ray luminosity, we estimate that NGC 5252 is emitting at a fraction ~ 0.005 of L(Edd). In this sense, this active nucleus appears to be a quasar relic, now probably accreting at a low rate, rather than a low black hole mass counterpart of a QSO.Comment: Accepted for publication in A&

Angular Momentum Transfer in the Binary X-ray Pulsar GX 1+4

We describe three presentations relating to the X-ray pulsar GX 1+4 at a workshop on magnetic fields and accretion at the Astrophysical Theory Centre, Australian National University on 1998, November 12-13. Optical and X-ray spectroscopy indicate that GX 1+4 is seen through a cloud of gravitationaly bound matter. We discuss an unstable negative feedback mechanism (originally proposed by Kotani et al, 1999), based on X-ray heating of this matter which controls the accretion rate when the source is in a low X-ray luminosity state. A deep minimum lasting ~6 hours occurred during observations with the RXTE satellite over 1996, July 19-21. The shape of the X-ray pulses changed remarkably from before to after the minimum. These changes may be related to the transition from neutron star spin-down to spin-up which occurred at about the same time. Smoothed particle hydrodynamic simulations of the effect of adding matter with opposite angular momentum to an existing disc, show that it is possible for a number of concentric rings with alternating senses of rotation to co-exist in a disc. This could provide an explanation for the step-like changes in Pdot which are observed in GX 1+4. Changes at the inner boundary of the disc occur at the same timescale as that imposed at the outer boundary. Reversals of material torque on the neutron star occur at a minimum in L_X.Comment: 10 pages, 5 figures; accepted for publication by PAS