The part of a delay no orbit can imitate

A pulsar in a binary is a clock on an orbit, and its pulses arrive early and late as it swings towards and away from the observer. That swing — the Rømer delay, the light-travel time across the orbit — is the whole of Newtonian pulsar timing, and it measures a projected semi-major axis, a period, an eccentricity and a periastron: everything about the orbit except its inclination and the two masses separately. General relativity adds small corrections that depend on the masses in different ways. One of them is the decay of the orbit, accumulating over decades; another is the precession of the periastron, Mercury’s anomaly scaled up by a factor of thirty thousand. A third happens once per orbit, at a definite moment, and has a shape.

When the pulsar is behind its companion, the pulses travel out of the system past the companion and through the curved space near it, and arrive late. The delay is the same effect that slows radar echoes grazing the Sun, measured there to a part in a hundred thousand by timing signals from a spacecraft behind the Sun. For a circular orbit,

$$\Delta_S = -2r\,\ln\!\big(1 - s\,\sin\Phi\big), \qquad r = \frac{G m_c}{c^3}, \qquad s = \sin i,$$

with $\Phi$ the orbital phase from the ascending node, $m_c$ the companion’s mass and $i$ the inclination of the orbit. The two parameters are exactly the two numbers the Rømer delay cannot supply.

Two numbers in one curve #

The range $r$ is the companion’s mass in time units: $GM_\odot/c^3 = 4.925$ microseconds, so a half-solar-mass white dwarf has $r = 2.46$ µs. That sets the overall scale. The shape parameter $s$ sets how close the line of sight passes to the companion at conjunction, and it enters inside the logarithm: as $s \to 1$ the argument at conjunction goes to zero and the delay to infinity. For the orbit of PSR J1614−2230, inclined at 89.17°, $s = 0.99989$, and the pulses pass the white dwarf at a projected distance of about thirty times its radius. The delay rises to 48.5 µs in a spike a few degrees of orbital phase wide.

So the curve encodes two numbers in two different ways, one in its height and one in its sharpness. The sharpness has a practical cost. The spike’s width in orbital phase is of the order of the angle by which the orbit misses edge-on — under a degree at the tip for J1614−2230, a few degrees across its base — so a campaign that samples the orbit evenly spends almost all its time where the delay is smooth and uninformative. The measurement that weighed that pulsar was made by observing it continuously through a conjunction, arranging the telescope’s schedule around a few hours of an eight-day orbit. At 70° the same companion gives a delay only a third as tall and so broad that it spans half the orbit. The mass is still in it; the question is whether it can be got out.

Why a logarithm #

The logarithm is not decoration, and where it comes from explains the spike. A signal crossing a region of gravitational potential $\phi$ runs slow, in the coordinate time of a distant observer, by a fraction $2|\phi|/c^2$ — half from the slowing of clocks, half from the stretching of space along the path. For a companion of mass $m_c$ the potential falls as $1/d$, so the extra time accumulated along a straight path is the integral of $2Gm_c/(c^3 d)$ along the path, and an integral of one over the distance, taken along a line that passes a point, grows as the logarithm of the path’s length divided by its distance of closest approach.

The path length is set by the orbit and does not change much; the closest approach changes enormously. At superior conjunction the pulses pass the companion at a distance equal to the orbital separation times $\cos i$, and as the inclination approaches ninety degrees that distance approaches zero and the logarithm grows without limit — in practice until the pulses would graze the companion’s surface or pass through a region where the approximation of a straight path fails. Away from conjunction the pulses leave the system on the pulsar’s side and the delay is small and slowly varying. The spike is the logarithm of a distance that the inclination can make arbitrarily small.

This is also why the same formula serves a radio pulse leaving a binary and a radar echo grazing the Sun, and why the solar measurement is so good: a signal from a spacecraft on the far side of the Sun can pass within a few solar radii of it, and a delay of more than two hundred microseconds builds up against a clock stable to nanoseconds.

What the orbit fit takes #

A delay drawn against orbital phase is not what a timing campaign measures. It measures the arrival times of pulses and fits a model to them, and the model already contains the orbit — which is itself a periodic delay at the orbital period, with a shape the fit adjusts freely.

For a nearly circular orbit the Rømer delay is

$$\Delta_R = x\left[\sin\Phi + \tfrac{\kappa}{2}\sin 2\Phi - \tfrac{\eta}{2}\cos 2\Phi\right],$$

with $x$ the projected semi-major axis in light-seconds and $\eta$, $\kappa$ the two components of the eccentricity. Its constant part is absorbed into the arrival-time offset, its first harmonic into $x$ and the time of the ascending node, and its second harmonic into $\eta$ and $\kappa$. Any part of the Shapiro delay that looks like a constant, a first harmonic or a second harmonic is therefore absorbed with it, silently: the fit returns an orbit very slightly wrong and a Shapiro delay that is not there.

Nearly edge-on, most of the spike survives, because a spike a few degrees wide contains every harmonic of the orbital period with nearly equal weight and the fit can remove only three of them. At 70° the delay is broad and smooth, most of its power lies in the harmonics the fit absorbs, and what survives is a small ripple that begins with the third harmonic.

That surviving third harmonic has a closed form. Expanding the logarithm in a Fourier series gives terms whose amplitudes are powers of

$$\varsigma = \frac{s}{1 + \sqrt{1 - s^2}} = \tan\frac{i}{2},$$

and the third harmonic’s amplitude is $\tfrac43 h_3$, with

$$h_3 = r\,\varsigma^3.$$

This is the orthometric amplitude, introduced in 2010, and it is what a careful timing analysis now reports instead of $r$ and $s$ when the orbit is not close to edge-on — because $h_3$ is what the data actually constrain, and $r$ and $s$ separately are then almost perfectly correlated along a curve of constant $h_3$. The figure’s surviving curves are computed by fitting and removing the three absorbed harmonics directly, and their third harmonic agrees with $\tfrac43 h_3$ to better than three per cent at both inclinations.

The cube of the half-angle #

The surviving amplitude falls with inclination as the cube of the tangent of half the angle, which is steeper than intuition expects.

For a half-solar-mass companion $h_3$ is 2.36 µs edge-on — the tangent of forty-five degrees is one — and falls to 0.47 at 60°, 0.12 at 40°. A neutron-star companion of 1.2 solar masses gives two and a half times more at every angle; a light white dwarf of 0.2 gives less than half. Against a timing precision of 0.1 µs, the level at which the most stable millisecond pulsars are timed over an hour’s observation, the half-solar-mass companion becomes detectable above about 53°. Against a microsecond, $h_3$ alone never reaches three times the precision at any inclination, and the delay becomes detectable only above about 75°, where the spike’s higher harmonics add enough to what survives to carry it over.

This has one clean consequence. The delay’s full height grows without limit as the orbit approaches edge-on, because of the logarithm; the part that can be measured does not, because $\varsigma$ saturates at one. An edge-on orbit is not infinitely better than a nearly edge-on one; it is better by the higher harmonics the spike carries, and those are what separate $r$ from $s$. Below about 80° the harmonics above the third are too small to measure and only the combination $h_3$ is returned; above it, the spike’s shape resolves the two parameters separately.

A neutron star weighed at two solar masses #

When both parameters are measured, the pulsar’s own mass follows from the orbit.

The Rømer delay gives the mass function,

$$f = \frac{(m_c \sin i)^3}{(m_p + m_c)^2} = \frac{4\pi^2 x^3}{T_\odot P_b^2},$$

which for J1614−2230 is 0.0205 solar masses: one equation in three unknowns. In the plane of the two masses it is a family of curves, one for each inclination. The shape of the Shapiro delay picks the curve, $\sin i = 0.999894$; its height gives the companion’s mass, $0.500 \pm 0.006$ solar masses; and the two cross at a pulsar mass of 1.97 solar masses. Had the orbit been inclined at 60° and nothing else changed, the same mass function would have made the pulsar 1.49 solar masses — an ordinary neutron star — and nobody would have known which.

The result, published in 2010, was the first precise measurement of a neutron star near two solar masses, and it was a crossing of two curves in this diagram. Every equation of state for matter at nuclear density predicts a maximum mass, and every one whose maximum lay below about 1.9 solar masses was excluded by it, including most models in which the core of a neutron star contains free hyperons or deconfined quarks at modest densities. A later pulsar with a Shapiro delay, heavier still at about 2.08 solar masses, tightened the bound. These are the heaviest well-measured neutron stars, and both were weighed by the delay their companions’ gravity put in their pulses — for a white dwarf, far below the limit on a cold star held up by electrons, whose own mass is part of the measurement.

A mass the orbital period had already predicted #

The companion of a millisecond pulsar is usually a white dwarf, and for the lightest of them the Shapiro delay tests a prediction made before any was measured. A millisecond pulsar is spun up by accreting matter from its companion, and the companion that fed it was a low-mass red giant overflowing the surface beyond which its gas belonged to the neutron star. A red giant’s radius is set by the mass of its degenerate helium core, almost independently of the envelope around it; the orbit’s size is set by the giant’s radius, because the giant filled its Roche lobe; and when the envelope was used up, the core was left as a white dwarf in an orbit whose period records the radius the giant had.

So the orbital period of a pulsar with a helium white-dwarf companion predicts the white dwarf’s mass: about 0.17 solar masses for an orbit of a day, rising to about 0.3 for an orbit of a hundred days, with a spread from the giant’s initial composition. The relation was worked out from stellar evolution in the 1990s. Where Shapiro delays have since measured the companions’ masses directly — for pulsars with orbits from under two days to several weeks — they fall on or near it, which is a test of the theory of red-giant cores carried out with the gravitational delay of radio pulses.

J1614−2230’s companion does not fit that relation, and its mass is part of the reason: at half a solar mass, far above what a giant of its orbital period could leave, it is a carbon–oxygen white dwarf, the remnant of a more massive star that transferred mass on a shorter timescale. The pulsar’s unusual mass and its companion’s unusual mass are a single history, and the Shapiro delay measured both.

Rare because of the clocks, not the angles #

It is often said that Shapiro-delay masses are rare because the geometry is rare: the orbit has to be nearly edge-on, and few are. The cube law suggests something different, and the calculation can be done.

For orbits oriented at random the cosine of the inclination is uniformly distributed, so the fraction of binaries above any threshold inclination is the cosine of that threshold. Judged on everything that survives the orbit fit, a half-solar-mass companion is weighable in 71 per cent of systems at a timing precision of 0.1 µs and in a quarter of them at 1 µs. At the precision the best millisecond pulsars reach, most binaries are inclined steeply enough. What makes the measurement rare is the precision itself: an ordinary pulsar, spinning a few times a second with a broad, noisy pulse, can be timed only to tens or hundreds of microseconds, and at ten microseconds no white-dwarf companion at any inclination leaves a detectable delay.

The statement that measurability is a question of geometry rather than of precision, made in an account of what pulsar timing fits for, is right about the separation of $r$ and $s$ — that genuinely needs an orbit within ten or twenty degrees of edge-on — and wrong about detection, which needs only a clock good enough. The distinction matters for what comes next. A pulsar timing array times several dozen millisecond pulsars to a tenth of a microsecond for decades to detect the background of gravitational waves, and the binaries among them are being weighed as a by-product: the population of Shapiro-delay masses has grown from a handful to a few dozen as the timing improved, not as more edge-on orbits were found.

When there are more curves than unknowns #

In a binary of two neutron stars the delay is one of several relativistic effects measurable in the same data, each a different function of the same two masses. The periastron advances at a rate set by the total mass; the pulsar’s clock runs slow and fast around an eccentric orbit by an amount set by both; the orbit shrinks at a rate set by their product and their sum; and the Shapiro delay gives the companion’s mass and the inclination. In the plane of the two masses each measured quantity is a curve, and general relativity predicts that every curve passes through one point.

For the system in which both neutron stars are pulsars, discovered in 2003 and seen within a degree of edge-on, six such curves have been measured, including the Shapiro shape parameter to a few parts in a hundred thousand and the ratio of the two masses directly from the two orbits. They meet at one point to within their errors, and the Shapiro delay’s shape agrees with the value the other parameters predict to about five parts in ten thousand — the most precise test of the theory in a strong field that pulsars have given. The companion there is itself a clock, and its own pulses are eclipsed for about thirty seconds at each conjunction by the magnetosphere of the pulsar in front, which is another geometric measurement of the same inclination, made with a different physics entirely.

What the figures leave out #

The orbits are circular, and the Shapiro delay of an eccentric orbit depends on the periastron’s orientation as well; the harmonic analysis carries over but the absorbed terms mix differently. The detection criterion — a surviving half-range above three times the precision — is a rule of thumb: a real campaign has hundreds or thousands of arrival times, spread unevenly over the orbital phase, and its sensitivity to a spike near conjunction depends on how densely conjunction was sampled, which is why campaigns are scheduled around it. The random-orientation fractions ignore selection: pulsars are found by surveys with their own biases, and binaries whose companions eclipse or distort the pulses near conjunction lose exactly the phases that matter. And the masses drawn for J1614−2230 are the values of its discovery paper; continued timing has since moved them by a few per cent, the pulsar slightly lighter, without changing what the crossing excluded.

Still open: the upper end of the neutron-star mass #

The heaviest neutron stars weighed by their companions’ delays sit a little above two solar masses. Heavier ones would move the bound on the equation of state further; the theoretical maximum for the stiffest plausible nuclear matter is about 2.3 to 2.5 solar masses, and a black hole’s lightest mass formed by stellar collapse is not known to within a factor of two. Whether any neutron star exists between 2.1 and 2.5 solar masses, and what separates the heaviest neutron star from the lightest black hole, are questions a single edge-on binary with a nanosecond-level clock could answer — by drawing two curves on this diagram and letting them cross.