Post-Roche Accretion

9. X-RAY BINARIES 152 Putting all of this together, we can compute the characteristic timescale τ_exp = h_s/R˙ over which ˙M increases by a factor of e. A plot of this is shown in figure 9.1.

Note that low mass stars have a much easier time expanding, both because they have lower thermal content and because they both can be and have to be much closer to the pulsar to satisfy the expansion criteria.

9. X-RAY BINARIES 153

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=1 . 0 L

_¯

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=10 . 0 L

_¯

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=25 . 0 L

_¯

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=50 . 0 L

_¯

6.4 7.2 8.0 8.8 9.6 10.4 11.2 12.0 12.8

Lo g

hs/˙R

Figure 9.1: The vertical axis is logP in seconds, the horizontal axis is the com- panion mass M in solar masses, and the color represents the log of the expansion timescalehs/R˙ in seconds. The four different plots correspond to four different pulsar luminosities.

9. X-RAY BINARIES 154 If this radius falls within the Pulsar’s light cylinder it will bury the magnetic field⁵. This occurs when

M˙c= L_p

qωGMpc = 5×10¹³P_p,−3^1/2 L_p L

g/s, (9.41)

where Pp,−3 is the pulsar period measured in milliseconds. We may compute the thermal energy lost when this mass leaves the star at∼10⁴K. The result is roughly 3×10⁻⁸Lp. As the input heat is expected to be only a few orders of magnitude below L_p, this effect is negligible. Thus we expect the limiting factor in the accretion process to be that the heat coming from the pulsar is blocked above a certain ˙M.

Now at the accretion rate ˙Mc we may estimate the structure of the accretion disk which forms. The accretion luminosity is

L_acc = GM_pM˙

R₀/2 . (9.42)

We may equate this with the heat flux of the disk as a black body, giving T = GM_pM˙

πR₀³σ

!1/4

. (9.43)

If the disk is optically thin, then the temperature gradient in the vertical direction is negligible. We will assume that this is the case, and later demonstrate its consistency in the regimes of interest. The remaining structural equations which must be solved

5There is some evidence that the actual radius to compare to is smaller than the light cylinder radius by a factor of 20 or so (Unal Ertan. “Inner disk radius, accretion and the propeller effect in the spin-down phase of neutron stars”. In: []. eprint: http://arxiv.org/pdf/1504.03996v1.pdf).

As this work is only suggestive, we proceed with the currently accepted model. If it turns out that a smaller radius is necessary, the critical accretion rates and associated luminosities will be reduced, which would mean a higher disk timescale and hence more type 1 cycles, as will be explained in subsequent sections. If the increase in timescale is sufficient, it could even allow asteroids and other similar objects to form in the disk, providing an explanation for some of the timing noise in pulsars with known companions.

9. X-RAY BINARIES 155 are⁶

h_s = v_sR^3/2₀

qGM_p (9.44)

v_s² = P

ρ (9.45)

αv_sh_sΣ = M˙

3πf, (9.46)

where α is a dimensionless parameter less than unity relating the viscosity of the disk to the product of h_s and v_s, and f is a dimensionless parameter equal to (1−^qR_inner/R₀)^1/4 ≈1. Solving for Σ yields

Σ≈ 2 ˙M m_p 3αk_BTP

s M

2 +M, (9.47)

where M is measured in solar masses. Plugging in ˙M = 10¹³g/s, P ∼ 10⁴s, and α >10⁻² yields Σ<2g/cm². Low-temperature opacities tend towards ∼1cm²/g, so for this ˙M the optically-thin assumption is valid. The worst case scenario for this assumption while still keeping ˙M sub-critical occurs whenT ∼10³K. For hotter disks, the opacity drops off by several orders of magnitude⁷. When T is 10³K, Σ is between 0.2g/cm² and 20g/cm², depending on the chosen value of α. Here τ can be as great as 10, so taking the system to be optically thin is perhaps not a good assumption. On the other hand, the ratio of the disk interior temperature to the surface temperature goes as the optical depth to the one-fourth power, and this ratio is the resulting error in the scale height and squared sound speed, so even an optical depth of 10 does not incur error in Σ greater than the existing error due to the uncertainty in α. Thus we will proceed with the optically-thin assumption.

Now the radial velocity of the accreting material is determined by the timescale over which viscosity dissipates angular momentum. This is given by⁸

τ_disk = R²₀

ν = R²₀

αh_sv₀ = R₀ α

s m_p

k_BT ∼3×10⁵sR^5/8₀ M˙_13.7^−1/8, (9.48)

6T. Padmanabhan. Theoretical Astrophysics. Vol. 2. ISBN: 978-0521566315. Cambridge University Press, 2001. Chap. 6.

7Jason W. Ferguson et al. “Low-Temperature Opacities”. In: The Astrophysical Journal 623.1 (2005), p. 585. url: http://stacks.iop.org/0004-637X/623/i=1/a=585.

8D. Lynden-Bell and J. E. Pringle. “The evolution of viscous discs and the origin of the nebular variables.” In: Monthly Notices of the Royal Astronomical Society168 (Sept. 1974), pp. 603–637.

9. X-RAY BINARIES 156 where R₀ is measured in solar radii. At the critical accretion rate, this is

τdisk,c= 3×10⁵sL^−1/8_p P_p,−3^−1/16R^5/8₀ , (9.49) where L_p is measured in solar luminosities, R₀ is in solar radii, and P is measured in seconds. This timescale may be viewed as the time over which material falling onto the outer edge of the disk travels to the inner edge when the accretion rate is near the critical value. We see that in most cases this is quite short, only of order one hundred orbits.

It is extremely important to note in this analysis that the pulsar field only turns off when the mass loss rateon the inner edge of the diskreaches the critical value.

In the event that the disk forms quickly relative toτ_exp, this ˙M is the same as the ˙M which fell onto the disk a time τdisk earlier, so there is a time delay associated with waiting for the material to reach the pulsar. This has two key impacts on our system.

First, it introduces the possibility of limit-cycles by building in a characteristic delay timescale, and second it allows the mass loss rate to continue to grow after ˙Mc has been reached at the companion. When the disk timescale is not too much longer than the expansion timescale, the typical overshoot in mass loss associated with this delay is

∆ ln ˙M = τ_disk τexp

. (9.50)

We can plot this using our numerical results forτ_exp. In figure 9.2 we have done this for a variety of L_p values with P_p = 10⁻³s. Examining the figure, we see in all cases that the growth is negligible, so the limit-cycle possibility is the key impact of the disk clearing time.

Of course, in the event that the disk forms slowly, the disk forming time may become the relevant parameter. In this case, we expect

∆ ln ˙M = τ_spread

τ_exp , (9.51)

where τspread is the time the disk takes to form and spread once the companion star overflows the Roche radius. We will consider this timescale in more detail in later sections.

9. X-RAY BINARIES 157

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=1 . 0 L

_¯

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=10 . 0 L

_¯

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=25 . 0 L

_¯

0.2 0.4 0.6 0.8 1.0 1.2 M

3.6 3.8 4.0 4.2 4.4 4.6 4.8

Log P

L

_p

=50 . 0 L

_¯

7.2 6.4 5.6 4.8 4.0 3.2 2.4 1.6

Lo g

τdisk/τexp

Figure 9.2: The vertical axis is logP in seconds, the horizontal axis is the companion massM in solar masses, and the color represents the log ofτ_disk/τ_exp. The four plots correspond to different pulsar luminosities.

9. X-RAY BINARIES 158