By Dinesh Kandel
When a supernova results in a neutron star (NS), the material that remains is fated to reach one of the most bizarre end-states of the stellar life cycle, rivaled only by black holes in weirdness. Fusion normally keeps the core of stars stably pressure-supported. However, after a sufficiently massive star exhausts its variety of nuclear fusion fuels, this pressure drops, initiating a supernova explosion and core implosion. Here the core suddenly collapses in on itself until the protons and electrons are forced to merge, so all that is left is a hard-packed, city-sized ball of nearly pure neutrons, a NS. This makes NSs the densest material objects known, cramming more matter than our entire Sun contains into a New York city-sized sphere, about a millionth of a billionth times smaller in volume than the Sun.
This highly compressed material is called degenerate matter, which is so dense it must be described using the laws of quantum mechanics. As a result, NSs should have a unique relationship between mass and radius, determined by an “equation of state” (EoS). An EoS is a thermodynamic description of the entwined properties of, for example, temperature, pressure, and volume. The steam in a pressure cooker can be described by an EoS; so can the interior of a star.
For NS-like conditions only density matters, which results in a density-dependent EoS, P(⍴). Thus far, because their material is in an exotic state not found anywhere on Earth, the EoS for NSs is not known or even well-constrained. To date, several candidate EoSs have been theoretically proposed based on different assumptions about how matter behaves at high density. These EoSs predict different values for the maximum mass an NS could have—thus the discovery of a very heavy NS could provide a significant constraint on the NS EoS, which will improve its ability to describe the behavior of matter under these extreme conditions.
This is where the studies that I and KIPAC professor Roger Romani have made of Black Widow systems enter the story. Black Widows are a class of binary system consisting of a rapidly spinning NS (called a pulsar) which is steadily devouring its lower-mass visible companion (these systems have been discussed before by Romani in a post here in September 2014). Because pulsars in these systems are believed to have been highly spun up due to accreting significant amounts of matter in the past, there are some good reasons to suspect that these systems potentially contain very massive NSs—massive-even-for-NSs-massive, that is.
In a Black Widow system, the extreme irradiation from the pulsar is slowly evaporating its companion. Such a system emits in radio, optical, X-ray, and gamma rays, revealing the rich physical phenomena in play. The optical light, for example, is emitted by the thermally heated companion. Because of the irradiation from the pulsar, the companion’s side (which permanently faces the pulsar due to tidal locking) is heated to a very high temperature, whereas the side opposite to the pulsar is relatively cool. (See Fig. 1, below, for a schematic of the temperature distribution of the companion star.)
Because of this temperature difference, the brightness variation of the star over time, called its lightcurve, exhibits periodic modulation. Proper modeling of this modulation gives information about the geometry of the binary system, including the inclination of the binary system relative to our line of sight to it. This information, combined with the radial velocity measurement of the system obtained through spectroscopic observations, provides a measurement of the mass of the system.
In our recent study of Black Widow system PSR J1810+1744, we discovered one of the most massive NSs known. The companion star in this system is heated to a balmy temperature of roughly 13,000 K on the pulsar-facing side, while it is at only 3,500 K on the opposite side, creating an extreme temperature asymmetry between the two sides. Because of this asymmetry, the companion must redistribute heat through surface winds, as is confirmed from the modeling of the system’s optical light. Moreover, the lightcurve modeling shows that the companion has locally heated spots on the surface, possibly due to charged particles being magnetically channeled to the surface of the companion (so generally these spots would be located at the magnetic poles of the star).
In addition, the NS and companion star form a binary system that results in a Roche lobe geometry. The Roche lobe is the region around each star in a binary system within which any material pulled off of each star is still gravitationally bound to it. These lobes are approximately teardrop-shaped, with the sharp point of the teardrop of the lobe around the companion star, also called the “nose,” pointing toward the pulsar. For a system where the companion star fully fills its Roche lobe, the surface gravity at the nose of the star is much smaller than on the rest of the star surface. This leads to an effective decrease in the temperature at the nose, due to a phenomenon called gravity darkening.
Furthermore, the pulsar-facing side of the star has what’s called a “radiative photosphere” because of the high temperature due again to irradiation from the pulsar. This accentuates the nose’s relative coolness compared to surrounding regions.
In effect, these two characteristics of the companion—the radiative photosphere and filled Roche lobe—greatly reduce the temperature of the nose region. Until our study, the importance of this phenomenon was not fully understood for irradiated stars. Our study shows that if these physical phenomena in the companion star are not included in the modeling, it not only leads to a worse statistical fit to the data but also results in an inferred NS mass of over three solar masses, which is precluded by other constraints on dense matter physics.
This happens because the mass estimation of a NS in a binary system involves two steps: determining the viewing inclination angle of the system through modeling of its observed lightcurve, and determining the line-of-sight projected center of mass (CoM) velocity of the companion star through radial velocity modeling.
From our method, mass is directly proportional to the cube of CoM velocity, and inversely proportional to the cube of the sine of the inclination angle, meaning that a lower inclination would mean a higher mass. (Ninety degrees is edge-on, zero degrees is face-on. See Fig. 2, below.)
The effect of gravity darkening is that it makes the nose temperature cooler. However, if that phenomenon is not considered, then the model tries to mimic it by lowering the inclination (as the viewing angle decreases we start seeing more of the poles, which are cooler). This decrease in inclination thus causes the estimated mass to be higher for simplistic models that do not take gravity darkening into account.
Our work suggests that the NS in PSR J1810+1744 has a mass of 2.13 +/- 0.04. This is almost as high as the recently discovered PSR J0740+6620 of mass 2.14 +/- 0.10, but our result is more precise (i.e., has much better statistical significance). This is an exciting result, which immediately helps us rule out several EoSs that have been theoretically proposed, as shown in Fig. 4.
To help explain Fig. 4, we begin by drawing the allowable mass range for NS J1810, shown as the hatched rectangular region centered at M = 2.2 Msun PSR J1810+1744 is the first object for which the 3σ lower bound for mass exceeds two solar masses, hence providing a high constraint on the NS EoS. In fact, by combining the mass measurements of all heavy NSs measured so far, we can constrain the 3σ lower bound of the maximum NS mass to be about 2.12 Msun.
What this tells us: Note how some of the colored curves in Fig. 4 do not (or just barely) intersect the rectangle denoting our pulsar’s possible mass range. The curves represent EoSs based on curious theoretical states of matter like quark-gluon plasma, hyperon matter, strange matter, and so on, but the figure shows that those EoSs don’t describe J1810. Instead, the simplest situation possible, neutrons packed tightly together with very little elbow room to make anything else, seems the most likely scenario.
Another way of saying this is if a neutron star reaches a high enough mass, the internals must be stiff enough to hold up the outer shells, meaning "hard-packed neutrons" is the only scenario allowed. There is no room for more hypothetical states of matter with exotic particles dancing about inside.
We want more data than one pulsar can provide, though, so now we are searching for even heavier Black Widow candidates to lead us to more insights into how nuclear matter behaves in these extreme states.
To be continued….