Critical Accretion Dynamics

9. X-RAY BINARIES 158

9. X-RAY BINARIES 159 and

V = 1

γ₀^qC(∇_rad − ∇_ad)

, (9.57)

where A, C, γ0, and∇ad are parameters independent of L and where V is typically within an order of magnitude of unity. Implicit differentiation of the second equation yields

6A

V y²∂_Ly−2A

V²y³∂_LV + 2y∂_Ly+y∂_LV +V ∂_Ly= 0, (9.58) and hence

∂_Ly=

2A

V²y³∂_LV +y∂_LV

V + 2y+ ^6A_V y² =− V

2(∇_rad− ∇_ad)

! _2A

V²y³ +y

V + 2y+^6A_V y²∂_L∇_rad. (9.59) This may be used to compute ∂_L∇as

∂L∇=y(y+V)∂L∇rad+ (∇rad − ∇ad)2y∂Ly+ (∇rad− ∇ad) (V ∂Ly+y∂LV) (9.60)

= y(y+V)−

2A V²y³+y V + 2y+^6A_V y²

"

V y+V² 2

#

− yV 2

!

∂_L∇_rad (9.61)

= y²(V + 2y)(V + 2Ay) V²+ 2V y+ 6Ay²

!

∂_L∇_rad. (9.62)

Now (∇_rad− ∇_ad)y(y+V) is a measure of the superadiabaticity of the convection.

Denote this by P. Then in efficient convection we expectP 1, and hencey(y+V) 1, as ∇_rad− ∇_ad is typically at least of order 0.1 in convection zones. Thus y1, so

∂lnLln∇ ∼y²∂lnLln∇_rad. (9.63) We can relate y to vc as¹¹

v_c=yv_s

s∇rad− ∇ad

8 ∼ 1

3yv_s∇^1/2_rad. (9.64) The ratio of the time dt over which the thermal structure in a layer adjusts to the thickness dr of the layer is

v_adj = F_e

ρc_pT∆ (∇_ad) = F_i

ρc_pT y²∇_rad = v³_c

v_s²y²∇_rad = 1

30yv_s∇^1/2_rad = v_c

10. (9.65)

11Ibid.

9. X-RAY BINARIES 160 whereas the corresponding rate for eddy adjustment is justv_c. Thus the true timescale over which the convection zone "notices" that the heating has been turned off is ten times the eddy timescale. The change in temperature is just

∆T = ∆L

4πR²ρv_adjc_p. (9.66)

The resulting rate at which R changes is R˙ =vadj

∆T

T = ∆L

4πR²ρc_pT = ∆L

10πR²P, (9.67)

where here ρ is evaluated at the location of the advancing flux adjustment wave and

∆L isL_e+L_i −L_surface. Using P ∼exp(v_adjt/l), we may write R˙ =v_adj∆T

T = ∆Le^−vct/10

4πR²P₀ , (9.68)

where P₀ is the pressure at the shallowest point in the convection zone above the ionization zone. When the adjustment wave reaches the base of the convection zone, R˙ approaches the negative of the expansion rate.

We now turn to the shallow case. When the external illumination is turned off the upper envelope temperature drops due to∂_mL−ε being nonzero in the heating zone.

Simultaneously, the convection zone adjusts to provide the flux needed to match the surface temperature. The boundary condition which reconciles these two is that the convection zone maintains an adiabatic gradient, and so deeper than the ionization zone its temperature must be effectively unchanged. The code we have used to compute the various expansion plots shown previously has a module which computes this process, once more using binary search. Here the objective is to minimize the deviation in the convection zone temperature in the parts deeper than the ionization zone from the heated value, and the free parameter is the surface temperature. The resulting stellar structure allows us to compute the extent of the fast contraction, occurring on a thermal timescale for the heating zone, for each heated stellar model.

The results of this calculation are shown in figure 9.3. Note that this contraction does not just occur on the day side of the star. In cases where substantial bloating has occurred, the flux profile has been altered by winds, so turning off the illumination does precisely what would be expected from these calculations. The characteristic timescale for adjustment of the day-side surface region is just

τ_rad ∼ 4πR²ΣhcpT

∆L = Li

∆L

Σhcp

T³σ

!

= 10⁴ Li

(T /T)³∆Ls. (9.69)

9. X-RAY BINARIES 161

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

_¯

4 3 2 1 0 1 2 3

Lo g

∆R/hs

Figure 9.3: The vertical axis is logP in seconds, the horizontal axis is the companion mass M in solar masses, and the color represents the log of the ratio of the quick contraction length to the scale height. The four plots correspond to different pulsar luminosities.

9. X-RAY BINARIES 162

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

_¯

4.0 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2

Lo g

τcontraction

Figure 9.4: The vertical axis is logP in seconds, the horizontal axis is the companion massM in solar masses, and the color represents the log of the contraction timescale.

The four plots correspond to different pulsar luminosities.

Now only the irradiated side can adjust this quickly: the other side, if radiative, will adjust on the wind timescale, typically an order of magnitude longer than the sound speed timescale. Thus 10⁶s is an upper bound on the timescale associated with radiative envelope adjustments. This is much faster than the eddy or convective thermal timescale, both at least four orders of magnitude larger, and so this will be the dominant timescale where applicable.

Putting it all together, we may compute the expected timescale over which the accretion rate drops by a factor of ten when the heating turns off. This is shown in figure 9.4. We see that for most stars, the time is quite short. For those which have the least sudden contraction, the timescale is longest, as expected. The divide is primarily one of mass, indicating that companion mass is the primary determining

9. X-RAY BINARIES 163 factor in the accretion response of the companion.

One thing worth noting is that the computed contraction timescales are, to leading order, the same as the expected expansion timescales if the pulsar is turned off and then back on. The ratio of the disk time to the contraction time can then provide a measure of ˙M overshoot, and is shown in figure 9.5.

9.5 Limit Cycles

Having now characterized the initial heating, post-Roche processes, and accretion disk dynamics, we now turn to the possibility of a limit cycle. The general picture is this:

1. The initial heat goes on until the star overflows its Roche-lobe.

2. The resulting accretion builds up a disk.

3. When ˙M = ˙M_c at the inside of the disk, a timeτ_disk after ˙M on the star reaches this value, the pulsar shuts off.

4. The accretion turns off as the companion cools.

5. The accretion disk clears after time τ_disk, after which the pulsar turns back on.

6. Accretion begins rapidly, as the radiative zone expands once more and begins to build a disk.

7. Time τ_disk later, the material reaches the pulsar and it turns off. The process then repeats.

There are four timescales which are potentially of interest for these cycles:

1. τ_disk - The time over which the equilibrium disk adjusts to perturbations.

2. τspread - The time over which a disk forms and spreads to the pulsar.

3. τ_M - The timescale over which ˙M changes by a factor of e prior to the limit cycle. Note that this is the same asτ_exp.

4. τ_M,L - The timescale over which ˙M changes by a factor of e inside the limit cycle. Note that this is what we have previously been callingτcontraction, as the

9. X-RAY BINARIES 164

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

_¯

2.0 1.5 1.0 0.5 0.0 0.5 1.0

Lo g

τdisk/τcontraction

Figure 9.5: The vertical axis is logP in seconds, the horizontal axis is the companion massM in solar masses, and the color represents the log of the ratio of the critical disk viscous timescale to the contraction timescale. The four plots correspond to different pulsar luminosities.

9. X-RAY BINARIES 165 corresponding timescale for expansion when the pulsar turns on is the same to leading order¹².

To investigate the properties of these limit cycles, we begin by computingτ_spread, as it is the only timescale of interest which we have not determined. To that end, suppose that we have a disk with inner radius R_i and outer radius equal to the companion orbital radius R₀. The area of the disk is then

A =πR²₀−R²_i, (9.70)

and the accretion luminosity is

L_a= GM_pM˙ Ra

, (9.71)

where

R_a≡ R₀ +R_i

2 (9.72)

is the mean radius. If the disk thermally equilibrates on timescales short relative to τ_spread, as we will verify is the case, then

T =

L_a 2Aσ

1/4

. (9.73)

We will later verify that Σ monotonically approaches its equilibrium value, such that our prior calculations showing that the surface and interior temperatures of equilibrium disks applies here as well. Assuming this for the moment, we find that the viscous timescale for the disk is

τ_visc = R²₀

ν = 3πR²₀v_sΣ

M f v˙ ₀ , (9.74)

where v0 is mean orbital speed, vs is the mean sound speed, Σ is the mean column density, and f is given by

f ≡ 1−

sRi

R₀

!^1/4

. (9.75)

In the limit where R₀−R_i R₀, we may write

R₀−R_i =εR₀. (9.76)

12There is a small discrepancy due to the changed thermal properties of the star, but this is irrelevant at the level of estimation being used here.

9. X-RAY BINARIES 166 In this regime,

f =

ε 2

1/4

(9.77)

A= 2πR²₀ε (9.78)

vs

v_s,0 =

A A₀

^−1/8

= (2ε)^−1/8, (9.79)

where v_s,0 is the equilibrium mean sound speed and v_s is the instantaneous mean sound speed. Using these approximations, as well as our expression in Eq. (9.48) for τdisk, we may expand τvisc as

τ_visc = 1

2^3+1/8τ_diskε^−11/8, (9.80)

whereτ_disk is the timescale for the equilibrium disk at this ˙M. Note that in performing this expansion, we made use of the fact that the mean orbital radius and mean sound speed both depend onε. The differential equation for the evolution ofε is therefore

∂_tε= 2^3+1/8τ_disk⁻¹ε^11/8, (9.81)

and hence

ε=

ε^−3/8₀ −3×2^1/8 t τ_disk

−8/3

. (9.82)

Setting this equal to unity, we see that τ_spread

τ_disk = ε^−3/8₀ −1

3×2^1/8 . (9.83)

This method of solution is justified by the fact thatτ_visc diverges at smallε, and hence we may focus on the time spent in that regime. Now typically we expect ε₀, which measures the initial disk width in units ofR₀, to be comparable to the atmospheric scale height of the companion. This is given by R_bv²_s/v²₀, so

ε₀ = R_b R₀

v_s v₀

2

= 0.46 M

M +M_p

!1/3

v_s v₀

2

. (9.84)

For an order of magnitude estimate, we note that M ∼ M, Mp ≈ 2M, and v_s ∼v₀/10, giving ε₀ ∼1/300. As a result, we may write simply

τ_spread τdisk

≈ 2 5





M M +Mp

!1/3v_s v0

2



−3/8

. (9.85)

9. X-RAY BINARIES 167 This tells us that the spreading time always exceeds the equilibrium disk viscous time.

The factor by which this occurs is typically of order a few, and so does not change the conclusion that the accretion rate does not change substantially from the critical value before the heating turns off.

To tie up loose ends, we now must verify some assumptions. First, consider the question of the monotonicity of Σ. We may write

Σ =˙ ∂

∂t

M_a A

= M˙

A −M_aA˙

A² = Σ (∂_tlnM_a−∂_tlnA), (9.86) where M_a is the disk mass. Now initially ∂_tlnA is roughly τ_spread⁻¹ and ∂_tlnM_a is infinite. Thus ˙Σ begins positive. Now Ma increases monotonically, so ∂tlnMa

decreases monotonically if M˙ is fixed. This decrease ends with a sharp drop to zero, coinciding with the time when the increase in A ends, as then the steady- state is achieved. Right before that time, Ma = M τ˙ _spread, so at all times before this ∂_tlnM_a ≥ τ_spread⁻¹ . As a result, Σ is monotonically increasing in time prior to equilibrium being established.

Next consider the question of thermal equilibration. The relevant dimensionless quantity of interest isM_ac_p∂_tT /L_a, where the time derivative is computed assuming thermal equilibrium. If the magnitude of this is less than unity then it is valid to assume thermal equilibrium. The rate at which T changes is given by

∂_tT =−1

4T ∂_tlnA =− T

4τ_spread. (9.87)

As a result, our dimensionless quantity is roughly M_av_s²/4L_a. The numerator is maximized in equilibrium, where Ma = M τ˙ spread, for v_s² ∝ T ∝ A^−1/4, whereas M_a= ΣAscales at least as A, as Σ increases monotonically inA. The denominator is maximized initially, asL_a ∝R⁻¹_a , so we may upper bound the quantity of interest by

τ_spreadv_s²R₀

4GM_p . (9.88)

For an order of magnitude estimate,R₀ ∼10¹¹cm,v²_s ≤3×10¹⁰cm²/s²,M_p ∼4×10³⁴g, 4G∼3×10⁻⁷cm³/g/s², so this quantity is at most 3×10⁻⁷τ_spreads⁻¹. From the last section, we know that τ_disk ∼3×10⁵s, so really we are interested in the quantity

2 50





M M +M_p

!1/3v_s v₀

2



−3/8

. (9.89)

9. X-RAY BINARIES 168 We may determine the maximum value of v₀/v_s by setting this equal to unity, giving a ratio of 2000. Typically v₀/v_s is at most 100, so we are safe in assuming thermal equilibrium.

Having determined that the spreading time exceeds the equilibrium viscous time, and having satisfied our various assumptions, we note that there are three possible limit cycle cases to consider:

1. τspread > τdisk > τM,L

2. τ_spread > τ_M,L > τ_disk 3. τ_M,L > τ_spread > τ_disk

We have not includedτ_M in these orderings because it is only relevant in determining how long the cycle takes to begin, and because it is typically much larger than the other three timescales.

Examining the timescales involved, we may classify regions of phase space into the different kinds of limit cycle. This is done in figure 9.6. The first thing to note about this plot is that as the pulsar luminosity increases, the low mass companions lose the possibility of a type 2 cycle. This results from the adjustment of the upper radiative zone increasing withL_p, thereby reducing τ_M,L. That the type 2 cycles also disappear as we go to higher M at fixedLp results from the deepening of the upper radiative layer as we approach the main sequence line.

The transition from type 1 to type 3 around 0.6M is due to the shrinkage and eventual disappearance of the heating-induced radiative zone. The radiative- convective transition is set by the condition that ∇_ad =∇_rad. In the upper layers of the star, this requires that the escaping luminosity be L_esc L_in, which makes the transition quite sharp. Additionally, the transition depends on the microphysics of opacity and ionization. These phenomena are often exponentially dependent on the thermal structure of the outer layers of the star, which further sharpens the change.

The transition from type 3 to type 2, and eventually to type 1 as M increases is just due to the convection zone shrinking, which reducesτ_M,L by reducing the relevant thermal mass. The thermal mass goes roughly as the convective base pressureP_f to the three-fifths power¹³. If we hold the period fixed, then to a good approximation R is fixed. As a result, the thermal mass just scales as M^6/5. We expect thatτ_M,L will be proportional to this. At the same time, the scale height is increasing, which counteracts this effect. We may compute the scaling as

h_s =R₀v_s² v₀

2

∼v_s² ∝T ∝ L^1/4

R^1/2 ∝ M^2.3/4

M^0.9/2 ∝M^0.13. (9.90)

13This is set by the adiabatic constantγ.

9. X-RAY BINARIES 169

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

_¯

Figure 9.6: The vertical axis is logP in seconds, the horizontal axis is the companion massM in solar masses, and the color represents the type of limit cycle. Blue is type 1, Green is type 2, Maroon is type 3. The four plots correspond to different pulsar luminosities.

9. X-RAY BINARIES 170 As this is a much lower power of M, we expect that τ_M,L∝h_s/M^6/5 will decrease as we move to higher mass.

Having established which kind of cycle occurs in which regime, it is worth con- trasting the various kinds of cycles. The first case corresponds to the fast expansion case. Based on figure 9.2, this case matters in a fairly wide regime. Here the limit cycle time is set by 2τ_spread+ 2τ_M,L. This may be seen by noting that when accretion begins, it takes time τ_spread for material to reach the pulsar. It then takes time τ_M,L for the accretion rate at the pulsar to reach the critical value, assuming that in the previous cycle it reached the critical value. It then takes timeτ_M,Lfor the accretion to halt. Finally, it takes time τ_spread for the disk to clear, allowing the process to begin again. The reasonτ_spread is relevant here is that the disk is never near equilibrium at any stage of the process. Given the orderings of timescales, we make at most a factor of two error by writing the full timescale as 2τ_spread. Naively, one might expect the corresponding overshoot in ˙M to be exp (τ_spread/τ_M,L). In practice this is not the case. To see why, note thatτ_spread and τ_disk both decrease with T, the latter as T^−1/2 and the former as T^−7/8. As a result, as T increases, the disk spreads faster, and so when τ_spread is considerably larger than τ_M,L, the increase in ˙M will be that required to set the relevant timescale connecting the outer and inner portions of the disk to τ_M,L. As that timescale is τ_spread in this case, and as T ∝ M˙ ^1/4 and τ_spread ∝ M˙ ⁻¹, we see that ˙M will increase by a factor of (τ_spread/τ_M,L)^15/8 over the critical value.

This typically involves just a few extra scale heights of motion, and so we do not expect the timescale τM,L to be off by too much from the actual timescale over which the mass loss rate adjusts, and at any rate the limit cycle timescale is dominated by τ_spread, so any corrections to the adjustment timescale effect are not relevant.

Now consider the second case. Here the limit cycle time is set by τ_spread+τ_disk+ 2τ_M,L. This may be seen by noting that when accretion begins, it takes time τ_spread for material to reach the pulsar. It then takes timeτ_M,L for the accretion rate at the pulsar to reach the critical value, assuming that in the previous cycle it reached the critical value. It then takes time τ_M,L for the accretion to halt. Finally, it takes time τ_disk for the disk to clear, allowing the process to begin again. To good approximation, given this ordering, we may simply say that the limit cycle takes time τ_spread, and thereby incur error of at most a factor of two given the large disparity betweenτ_spread andτ_disk. The overshoot is given by the same expression as in the first case, for once more the disk is out of equilibrium the entire time.

In the third case, the expansion is slow, and so the spreading time is irrelevant to the overshoot. The limit cycle time is once more set by τ_spread+τ_disk+ 2τ_M,L. Given the ordering of timescales, we may approximate this as 2τM,L, and thereby at most a 50% error. The overshoot in ˙M is exp (τ_disk/τ_M,L), for now τ_disk is the timescale

9. X-RAY BINARIES 171 mediating the delay between the companion and the accretion onto the pulsar.

In summary, then, we expect that there are three kinds of accretion disk limit cycles which are unique to these illuminated companion systems. These cycles are characterized by the relative orderings of the companion atmospheric timescale, the critical accretion disk formation timescale, and the equilibrium critical accretion disk viscous timescale. The cycles range in timescale from days to years, with on-off times typically measured in days. The key difference between the cycles is generally the process modulating them, either atmospheric effects or disk dynamics, as well as the scales of these effects. The luminosity of the accretion disk in each case is given approximately by the accretion luminosity of the mid-disk ring, and corresponds to a mass loss rate on the order of 10^14±1g/s. This puts the accretion disk radiation somewhere between the IR and soft X-Ray bands, depending on the precise system parameters. The luminosity of the accreting material at the pulsar is therefore of order 10^35±1erg/s. This radiation is expected to be mostly X-Rays, as is typical of accreting magnetic neutron stars.

9. X-RAY BINARIES 172

References

Brown, J. C. and C. B. Boyle. “An exploratory eccentric orbit ’Roche lobe’ overflow model for recurrent X-ray transients”. In: Astronomy and Astrophysics 141 (Dec.

1984), pp. 369–375 (cit. on pp. 145, 146).

Ferguson, Jason W. 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 (cit. on p. 155).

King, A. R. et al. “Mass Transfer Cycles in Close Binaries with Evolved Companions”.

In: The Astrophysical Journal 482 (June 1997), pp. 919–928. eprint: astro - ph/9701206 (cit. on p. 145).

Lynden-Bell, D. and J. E. Pringle. “The evolution of viscous discs and the origin of the nebular variables.” In:Monthly Notices of the Royal Astronomical Society 168 (Sept. 1974), pp. 603–637 (cit. on p. 155).

Paczyński, B. “Envelopes of Red Supergiants”. In: Acta Astronomica 19 (1969), p. 1 (cit. on pp. 158, 159).

— “Evolutionary Processes in Close Binary Systems”. In: Annual Review of Astron- omy and Astrophysics 9 (1971), p. 183. doi: 10.1146/annurev.aa.09.090171.

001151 (cit. on p. 146).

Padmanabhan, T.Theoretical Astrophysics. Vol. 2. ISBN: 978-0521566315. Cambridge University Press, 2001. Chap. 6 (cit. on p. 155).

Ritter, H., Z.-Y. Zhang, and U. Kolb. “Irradiation and mass transfer in low-mass compact binaries”. In:Astronomy and Astrophysics 360 (Aug. 2000), p. 969. eprint:

astro-ph/0005480 (cit. on p. 145).

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 (cit. on p. 154).