4 Formation of contact

In this section, we present results for 74 computed systems. Rather than providing a detailed description of the evolution of each system, we focus on the general trends of key properties as function of the major initial parameters. Table 3 provides the most important quantities for all computed systems. We focus on the various mechanisms which can drive massive close binary systems into contact.

 

 

Table 3: Characteristic quantities for all computed systems: system number, initial values of primary mass M₁, secondary mass M₂, and period P_i, and the mass transfer case. Bold values for the initial masses identify the component which ends its evolution first. Further columns are: the kind of contact experienced by the system (no contact = "-'', premature contact = "M'', reverse contact = "R'', q-contact = "Q'', and delayed contact = "D''; a lower case letter is used if the secondary fills less than 1.5 times its Roche lobe; systems Nos. 52 and 59 experience a second contact after a weak q-contact), the core helium mass fraction of the primary at the onset of the first mass transfer $Y_{\rm 1}$, and $M_{\rm 1}^{\prime }$, $M_{\rm 2}^{\prime }$, $L_{\rm 1}^{\prime }$, $L_{\rm 2}^{\prime }$, $R^\prime _1$, $R^\prime _2$, $P^{\prime }$ denote primary and secondary mass, luminosity, effective temperature, radius and orbital period after the Case A/AB or Case B mass transfer, at a time when the core helium mass fraction of the primary has decreased to 0.8 due to helium burning. $\dot{M}_\mathrm{max}$ is the maximum mass transfer rate (lower limits are give for contact systems, values in italic are uncertain) and $\Delta M$ is the amount of mass transfered by the time the maximum mass transfer rate or contact is reached. $\beta$ designates the fraction of the transfered matter that is accreted, assuming no further accretion after the secondary overfills its Roche lobe more than 1.5 times

No.	M1	M2	Pi	case		Y</I>1	$M^\prime_\mathrm{1}$	$M^\prime_\mathrm{2}$	$L^\prime_\mathrm{1}$	$L^\prime_\mathrm{2}$	$R^\prime _1$	$R^\prime _2$	$P^{\prime }$	$\dot{M}_\mathrm{max}$	$\Delta M$	$\beta$
	$M_{\odot}$		d			%	$M_{\odot}$		log($L_{\odot}$)		$\,R_\odot$		d	$\frac{10^{-4} \,M_\odot}{\mathrm{yr}}$	$\,M_\odot$
1	12	11.5	2.5	A	-	87	1.42	21.5	2.98	4.99	0.3	16.5	218	0.71	1.8	1.00
2	12	11	3	B	-	98	2.44	20.4	3.93	4.74	0.8	7.2	54	4.0	7.3	1.00
3	12	11	6	B	-	98	2.38	20.4	3.82	4.79	0.6	6.8	115	10	7.3	1.00
4	12	11	10	B	-	98	--	--	--	--	--	--	--	17	7.5	1.00
5	12	11	16	B	-	98	--	--	--	--	--	--	--	25	7.5	1.00
6	12	11	22	B	-	98	--	--	--	--	--	--	--	28	7.6	1.00
7	12	11	30	B	D	98	--	--	--	--	--	--	--	>33	6.5	0.68
8	12	11	40	B	D	98	--	--	--	--	--	--	--	>33	5.7	0.60
9	12	10.5	2	A	M	75	--	--	--	--	--	--	--	0.67	1.5	0.70
10	12	10.5	2.5	A	-	87	1.42	20.7	2.97	4.87	0.3	7.9	190	0.93	2.0	1.00
11	12	10	2	A	M	76	--	--	--	--	--	--	--	0.82	1.6	0.68
12	12	10	16	B	-	98	2.43	19.4	3.81	4.71	0.6	6.6	255	22	7.0	1.00
13	12	10	22	B	D	98	--	--	--	--	--	--	--	>28	6.9	0.73
14	12	9.5	2	A	-	76	1.29	19.4	2.93	5.00	0.4	32.3e	171	1.8	2.0	1.00
15	12	9.5	2.5	A	-	88	1.41	19.7	2.96	4.79	0.3	7.5	166	1.2	2.0	1.00
16	12	9.5	3.5	B	-	98	2.45	18.9	3.94	4.63	0.8	6.3	50	4.0	2.9	1.00
17a	12	9	2	A	R	76	1.15	19.4	2.67	4.87	0.3	9.0	214	1.2	1.7	1.00
18	12	9	3.5	B	-	98	2.32	18.5	3.74	4.63	0.6	6.5	53	6.5	3.4	1.00
19	12	9	6	B	-	98	2.36	18.5	3.81	4.63	0.6	6.2	87	9.9	7.4	1.00
20	12	9	10	B	-	98	--	--	--	--	--	--	--	15	7.0	1.00
21	12	9	16	B	d	98	--	--	--	--	--	--	--	28	7.0	1.00
22	12	9	22	B	D	98	--	--	--	--	--	--	--	>29	5.9	0.62
23	12	8.5	2.5	A	-	88	1.43	18.8	2.97	4.72	0.3	6.8	135	1.6	2.1	1.00
24	12	8	2	A	-	77	1.12	18.5	2.64	4.84	0.3	8.5	189	3.0	2.3	1.00
25c	12	8	2.5	A	R	87	1.66	17.9	3.18	4.68	0.4	6.5	81.5	3.6	2.5	1.00
26	12	8	3.5	B	q	98	--	--	--	--	--	--	--	7.3	3.0	1.00
27	12	8	6	B	-	98	2.37	17.5	3.79	4.55	0.6	5.8	72	8.9	3.5	1.00
28	12	8	10	B	-	98	--	--	--	--	--	--	--	15	6.8	1.00
29	12	8	16	B	D	98	--	--	--	--	--	--	--	>24	6.0	0.63
30	12	7.5	2	A	q	78	--	--	--	--	--	--	--	2.1	2.0	1.00
31	12	7.5	2.5	A	-	89	1.45	17.8	2.99	4.64	0.3	6.3	104	2.5	2.0	1.00
32	12	7.5	3.5	B	Q	98	--	--	--	--	--	--	--	>8.1	2.8	0.27
33	12	6.5	2	A	Q	78	--	--	--	--	--	--	--	>4.6	1.4	0.13
34	16	15.7	3.2	A	-	91	2.79	27.8	3.88	5.14	0.5	9.3	75	3.5	3.6	1.00
35	16	15.7	6	B	-	98	3.49	27.5	4.17	5.14	0.6	8.3	86	22	7.7	1.00
36	16	15	2.5	A	M	80	--	--	--	--	--	--	--	2.5	2.6	0.70
37	16	15	8	B	-	98	3.63	26.7	4.20	5.10	0.7	8.9	96	29	8.0	1.00

 

Table 3: continued

No.	M1	M2	P</I>i	case		Y</I>1	$M^\prime_\mathrm{1}$	$M^\prime_\mathrm{2}$	$L^\prime_\mathrm{1}$	$L^\prime_\mathrm{2}$	$R^\prime _1$	$R^\prime _2$	$P^{\prime }$	$\dot{M}_\mathrm{m}$	$\Delta M$	$\beta$
	$M_{\odot}$		d			%	$M_{\odot}$		log($L_{\odot}$)		$\,R_\odot$		d	$\frac{10^{-4} \,M_\odot}{\mathrm{yr}}$	$\,M_\odot$
38	16	15	9	B	d	98	3.64	26.7	4.22	5.10	0.7	8.1	107	32	8.1	1.00
39	16	15	15	B	D	98	--	--	--	--	--	--	--	>37	6.7	0.48
40	16	14	2	A	M	69	--	--	--	--	--	--	--	2.2	2.2	0.86
41a	16	14	2	A	-	68	2.12	27.0	3.68	5.21	0.6	11.0	110	1.2	2.1	1.00
42	16	14	2.5	A	-	80	2.39	26.5	3.78	5.17	0.6	11.0	97.5	3.1	2.9	1.00
43	16	14	3	A	-	87	2.52	26.6	3.75	5.10	0.5	9.0	101	2.0	2.8	1.00
44	16	14	6	B	-	98	3.60	25.8	4.19	5.04	0.6	7.8	67	24	7.9	1.00
45	16	14	9	B	D	98	3.64	25.7	4.21	5.04	0.7	7.6	98	31	8.2	0.64f
46	16	13	2	A	M	69	--	--	--	--	--	--	--	3.0	2.5	0.68
47	16	13	2.5	A	-	80	2.28	26.1	3.65	5.11	0.5	9.1	99	2.1	2.4	1.00
48c	16	13	2.5	A	-	79	2.47	25.6	3.77	5.10	0.6	7.6	78	4.3	3.0	1.00
49	16	13	3	A	-	88	2.64	25.3	3.84	5.07	0.5	9.1	64	5.2	3.5	1.00
50c	16	13	3	A	-	87	2.66	25.2	3.81	5.08	0.5	7.8	62,5	5.1	3.2	1.00
51	16	13	4	B	d	98	3.55	24.8	4.19	4.99	0.7	8.2	41	18	6.9	1.00
52	16	13	6	B	D	98	3.70	24.7	4.24	4.95	0.7	6.8	57	27	7.8	0.62f
53	16	12	1.5	A	M	54	--	--	--	--	--	--	--	3.5	2.4	0.71
54	16	12	1.7	A	M	61	--	--	--	--	--	--	--	3.7	2.5	0.75
55	16	12	2	A	-	70	2.07	24.8	3.57	5.17	0.5	11.8	93	4.2	2.8	1.00
56	16	12	4	B	d	98	3.55	23.9	4.17	4.93	0.6	7.0	37	18	6.3	1.00
57	16	12	6.5	B	D	98	--	--	--	--	--	--	--	26	7.7	0.38f
58	16	11	2	A	-	71	2.00	24.1	3.56	5.10	0.5	10.8	89	5.7	3.1	1.00
59c	16	11	2	A	qR	69	2.10	24.0	3.54	5.13	0.5	8.4	77	5.9	3.1	1.00
60	16	11	3	A	-	89	2.52	23.8	3.74	4.95	0.5	7.3	61	3.8	2.8	1.00
61	16	11	3.2	A	-	92	2.81	23.3	3.87	4.93	0.5	7.4	42	8.2	4.0	1.00
62	16	10	2.5	A	q	81	2.28	23.1	3.77	4.97	0.6	7.8	65	8.7	3.5	1.00
63	16	10	3	B	q	89	--	--	--	--	--	--	--	9.9	3.9	1.00
64	16	9	2.5	A	Q	82	--	--	--	--	--	--	--	12	3.4	0.16f
65	25	24	3.5	A	-	83	5.15	40.6	4.53	5.54	0.6	13.7	41	3.0	4.6	1.00
66a	25	24	3.5	A	-	83	5.21	40.6	4.54	5.54	0.6	12.4	40	3.0	4.2	1.00
67b	25	24	3.5	A	-	82	--d	40.5	--d	5.54	--d	11.2	--	3.0	4.6	1.00
68	25	24	5	B	d	98	7.28	39.2	4.85	5.45	0.8	11.9	31	30	8.2	1.00
69	25	24	9	B	D	98	--	--	--	--	--	--	--	>37	7.4	0.43
70	25	23	4	A	-	87	5.31	39.4	4.56	5.49	0.6	12.4	39	4.0	5.2	1.00
71	25	22	2.5	A	-	70	4.80	39.6	4.45	5.55	0.7	14.2	39	2.7	2.9	1.00
72c	25	22	2.5	A	-	68	4.68	38.9	4.44	5.62	0.7	10.5	39	5.3	3.6	1.00
73	25	19	4	A	-	88	5.26	35.9	4.55	5.38	0.7	8.7	30	6.3	4.9	1.00
74	25	16	4	A	-	89	5.22	33.3	4.54	5.29	0.6	8.8	24	8.8	5.0	1.00

^a $\alpha_\mathrm{sc} = 0.02$.
^b $\alpha_\mathrm{sc} = 0.04$.
^c Schwarzschild criterion instead of Ledoux criterion used.
^d Only secondary is computed but primary should not be very different from Nos. 65 and 66.
^e Secondary at ECHeB.
^f Treated as if $\beta = 1$.

Figure 5: Mass transfer rates as a function of secondary mass for Case B systems initially consisting of a 12$\,M_\odot$ and a 11$\,M_\odot$ star with initial periods in the range 6...40$\,$d (Nos. 3, 4, 5, 6, 7 and 8). The systems with 30 and 40 days initial period form contact, at which time we stop the calculation

4.1 Contact formation due to small or large periods

4.1.1 Case B

Case B systems with an initial mass ratio close to 1 and a rather short initial period (early Case B) are known to have a good chance to avoid contact during the Case B mass transfer (cf. Pols 1994). Our Table 3 contains 15 examples, i.e. systems Nos. 2...6, 12, 16, 18...20, 27, 28 (initial primary mass of $12\,M_\odot$), and Nos. 35, 37 and 44 (initial primary mass of $16\,M_\odot$). The fact that no such system with an initial primary mass of $25\,M_\odot$ could be found is discussed in Sect. 5.1. It is often assumed that late Case B systems, i.e. such with a relatively large initial period, evolve through a common envelope stage (Podsiadlowski et al. 1992; Vanbeveren 1998a,b,c). In the following, we will have a closer look to the transition region in between both extremes.

Figure 5 shows the mass transfer rates as a function of the secondary mass for Case B binaries with initial primary and secondary masses of 12$\,M_\odot$ and 11$\,M_\odot$, respectively, but different initial periods in the range 6...40$\,$d (Systems Nos. 3 to 8). Generally, initially wider systems develop larger mass transfer rates. However, it is interesting to consider the time dependence of the mass transfer rate of these systems in some detail. The first maximum in the mass transfer rate is related to the end of decrease of the orbital separation at mass ratio one. I.e., when the primary has become the less massive component in the system, the period and the Roche lobe of the primary begin to grow, and the increase of the mass transfer rate flattens. This happens with some time delay due to the finite thermal response time of the primary.

Later during the mass transfer, the orbit widens significantly. Therefore, the primary becomes more extended and its surface temperature decreases. In the wider systems considered in Fig. 5, the outer envelope of the primary becomes even convectively unstable. It is the drop of the adiabatic mass-radius exponent $\zeta_{\rm ad}$ (cf. Ritter 1988) which leads to another rise of the mass transfer rate even though the orbit widens. We designate this feature as delayed contact. Table 3 shows nine systems which evolve delayed contact. It is restricted to the Case B and has no counterpart in our Case A systems (cf. Sect. 4.1.2).

Figure 6: Radius of the secondary during mass transfer as a function of its mass, for the same systems as in Fig. 5. In the systems with 30 and 40 days initial period the large increase in radius leads to contact, at which time we stop the calculation

Figure 6 displays the radius evolution of the secondary stars of the systems considered in Fig. 5. Initially, the radii do not grow significantly, although it can be seen seen that larger radii are obtained for larger initial periods, i.e. for larger mass transfer rates. Later on, mass transfer rates in excess of $\sim$ $3\, 10^{-3}\, \,M_\odot~{\rm yr}^{-1}$ make the secondary swell enormously. This mass transfer rate corresponds roughly to $\dot M = R L / (G M)$, i.e. to the condition $M/ \dot M = G M^2 / (R L)$ for the secondary. In the two systems with initial periods of 30 and 40 days, the radius increases until the secondary also fills its Roche lobe and a contact system is formed. This effect marks the limit of contact free systems towards larger periods.

It is remarkable that approximately 5 $M_{\odot}$ for the 40 d binary (system No. 8) and 6 $M_{\odot}$ for the 30 d binary (system No. 7) are transfered before the contact occurs (cf. Fig. 6). From Fig. 6 we see that contact occurs the earlier (in terms of already transfered mass) the larger the initial system period is. I.e., there is a continuous transition towards very wide binaries with convective primaries, which evolve into contact at the very beginning of the mass transfer process (Podsiadlowski et al. 1992).

According to the estimate of Webbink (1984), which compares the orbital energy of the binary with the binding energy of the envelope, our System No. 7 does not merge during the Case B mass transfer. Instead, with an efficiency parameter of $\alpha_\mathrm{ce}=1$, it obtains a period of about 7.5 d in a common envelope evolution by expelling the remaining envelope of the primary of $\sim$ $3\,M_\odot$ from the system.

In summary, in delayed contact systems a conservative part of Case B mass transfer is followed by a non-conservative one, leading to an oscillation of the orbit: it first shrinks until the mass ratio is 1, then widens up to the end of the conservative part, and then shrinks again due to mass and angular momentum loss from the system. For our System No. 7, the parameter $\beta$ for the complete Case B mass transfer, i.e., the ratio of the amount of mass accreted by the secondary to the amount lost by the primary, is about $\beta \simeq 6\,M_\odot/ 9\,M_\odot\simeq 0.67$.

After the delayed contact and common envelope evolution, System No. 7 consists of a $\sim$ $20\,M_\odot$ main sequence star - the secondary - and a $\sim$ $2.4\,M_\odot$ helium star in a 7.5 d orbit. The further evolution (which we have not computed) depends on the post-common envelope period. For 7.5 d, the primary fills its Roche lobe as it expands to a helium giant, and transfers part of its helium envelope to the secondary in a Case BB mass transfer, similar to that discussed in Sect. 3.2 for system No. 37.

In contrast, the systems which avoid contact during the Case B mass transfer all together can often, even if the secondary evolves faster, avoid the reverse Case B mass transfer if the secondary star does not rejuvenate (cf. values of $T_{\rm eff,s}^{\prime}$ in Table 3). After the Case B mass transfer, their periods are typically larger than 50 d.

We find that in our most massive systems, with a primary mass of 25$\,M_\odot$, even early Case B mass transfer leads into contact, even for an initial mass ratios very close to 1 (cf. systems Nos. 68 and 69 in Table 3). In the two considered systems, large amounts of mass are transferred conservatively before contact occurs (8.2$\,M_\odot$and 7.4$\,M_\odot$; cf. Table 3), i.e., we have a delayed contact situation. In system No. 68, the contact is marginal, in system No. 69 it is not.

Figure 7: Secondary radius as a function of its mass during Case A mass transfer, for three systems starting out with a 16$\,M_\odot$ and a 12$\,M_\odot$ component, with initial periods in the range 1.5...2 d (Systems Nos. 53...55). In the two closer systems, the secondaries terminate core hydrogen burning during the slow phase of the Case A mass transfer. Their ensuing rapid expansion leads to contact. In the 2 d period system No. 55 (solid line), Case AB mass transfer starts (at $M_2\simeq 21\,M_\odot$) before the secondary has exhausted hydrogen in the core, and contact is avoided (see text)

4.1.2 Case A

Figure 7 demonstrates a completely different mechanism of contact formation operating in Case A systems, which occurs for short periods (cf. also Pols 1994). It shows the radius evolution of three secondaries in Case A systems which have identical initial parameters except the period (systems Nos. 53...55 in Table 3). All three secondaries evolve through the rapid part of the Case A mass transfer (cf. Sect. 3.1) without expanding significantly (which is different for initial mass ratios $q \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}0.65$; cf. Sect. 5.1). However, in the two closer systems, the mass transfer appears so early in the evolution of the primary that the now more massive secondary finishes core hydrogen burning first. It then starts to expand, but soon fills its Roche lobe and attempts a reverse Case B mass transfer (in systems Nos. 53 and 54). Thus, both stars now fill their Roche lobes and come into contact while the primary is still burning hydrogen in its core; we designate this as premature contact. The likely outcome of premature contact is a merger because the corresponding periods are too small for a successful common envelope ejection. This effect marks the lower period limit for contact free binaries.

In model No. 55, the primary ends core H-burning first. Therefore, Case AB mass transfer (cf. Sect. 3.1) sets in before the secondary expands strongly. This has two effects: it drives the two stars further apart - the period increases from 5.24 d at the beginning to 93 d at the end of Case AB mass transfer -, and it enhances the mixing of hydrogen into the core (cf. Braun & Langer 1995) and thus delays the evolution of the secondary. I.e., if the initial period exceeds a critical value, Case AB mass transfer starts before the secondary expands and premature contact can be avoided. If the secondary star does not rejuvenate, reverse Case B mass transfer is avoided as well. Curiously, the secondary star in our Case A system No. 1 finishes core hydrogen burning just after the end of the Case A mass transfer and before the beginning of the Case AB mass transfer. However, the accretion due to the Case AB mass overflow leads to a shrinkage of its radius, and contact is avoided (see also Fig. 11 below).

The lower period limit for contact free binaries depends sensitively on the convection criterion. As mentioned in Sect. 2, we compute stellar models using the Ledoux criterion for convection and semiconvection with an efficiency parameter of $\alpha_\mathrm{sc} = 0.01$. The semiconvective mixing efficiency - which is infinite in the case of the Schwarzschild criterion for convection - determines to a large extent whether an accreting core hydrogen burning star rejuvenates or not (Braun & Langer 1995). If it fails, which is the more likely for less efficient semiconvection and the more advanced the star is in its evolution when it starts to accrete, it retains an unusual structure, with a smaller helium core mass and a larger hydrogen envelope mass compared to single stars. As such a structure keeps the star relatively compact during all the advanced burning stages - i.e., it avoids the red supergiant stage (Braun & Langer 1995) - it will not fill its Roche lobe after core hydrogen burning. This is the situation in all our Case A systems which avoid premature contact and have initial mass ratios of $q \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}0.65$.

For models computed with the Schwarzschild criterion, the secondaries in Case A systems rejuvenate during or shortly after the rapid Case A mass transfer. This results in longer main sequence life times for the secondaries, and thus leads to less premature contact situations. I.e., the critical initial period for premature contact is shifted to smaller values. In this case, the secondaries all expand to red supergiants. Assuming the primary star is still present at that time, this is likely to initiate reverse Case B mass transfer. As the life time of a rejuvenating star is extended, the primaries have often finished their evolution by that time and the binary may even be disrupted (cf. Pols 1994). However, as the life time of a low mass helium star (i.e. the primary of such a system) can be comparable to the remaining hydrogen burning life time of the very massive secondary, reverse Case B mass transfer onto the low mass helium star may also occur. In this case, the helium star builds up a hydrogen envelope and swells to red giant dimensions. We designate the ensuing contact situation in Table 3 as reverse contact.

This scenario is illustrated by five of our Case A systems (Nos. 25, 48, 50, 59, and 72) which are computed using the Schwarzschild instead of the Ledoux criterion (marked "c'' in Table 3). Nos. 25 and 59, initially having a 12$\,M_\odot$ and a 16$\,M_\odot$ primary, a mass ratio of 0.67 and 0.69, and a period of 2.5 d and 2.0 d, respectively, avoid premature contact. However, they encounter reverse contact when the primaries have burnt 91% and 80% of their helium in the core. Both systems avoid contact altogether when they are computed with our standard convection physics, and evolve a reverse supernova order. I.e., an increase of the semiconvective mixing efficiency may reduce the number of systems with premature contact but increase the number with reverse contact. From system Nos. 25 we conclude that, were the Schwarzschild criterion correct, the majority of Case A binaries with 12 $M_{\odot}$ primaries would evolve into reverse contact. Comparing our models Nos. 58 and 59, which have identical initial conditions but vary in the treatment of convection, shows that also in systems with 16 $M_{\odot}$ primaries the lower period limit for contact free systems increases.

In the three other systems computed with the Schwarzschild criterion, systems Nos. 48, 50, and 72, the primaries explode as supernovae before the reverse mass transfer can occur. Two of these (Nos. 48 and 72), when computed with our standard convection physics, have a reverse supernova order (see systems Nos. 47 and 71 in Table 3), the other (No. 50) does not (cf. system No. 49 in Table 3).

Thus, the lower period limit for contact free evolution with the Schwarzschild criterion is expected between the lower period limit and the limit for reverse supernova order for Ledoux criterion and $\alpha_\mathrm{sc} = 0.01$. The one early case A system containing a 25 $M_{\odot}$ primary (No. 72) also shows no reverse mass transfer before the supernova explosion of the primary, despite being a system with reverse supernova order when computed with our standard convection physics. We conclude that for large masses, the amount of contact free case A systems does not change very much for different convection physics. On the contrary, at 12 $M_{\odot}$ primary mass and below no contact free case A system are expected at all if the Schwarzschild criterion is used.

4.2 Contact formation due to extreme mass ratios

4.2.1 Case B

While the previous two subsections concerned contact formation due to large (delayed contact) or small (premature and reverse contact) initial periods, we investigate here contact formation due to large or small initial mass ratios.

Figure 8 shows the mass transfer rate as function of the transferred amount of mass for three Case B systems with identical initial primary mass (12$\,M_\odot$) and period (6 d), but different initial secondary masses. For smaller initial mass ratios $q_{\rm i}=M_{2,{\rm i}} / M_{1,{\rm i}}$, the mass transfer rates are somewhat larger in the early part of the mass transfer, since the orbit shrinks more rapidly for smaller $q_{\rm i}$. For conservative evolution (also ignoring stellar winds), the minimum separation depends on initial separation $d_{\rm i}$ and mass ratio $q_{\rm i}$as $d_{\rm min} = d_{\rm i} \left( 4 q_{\rm i} \over (q_{\rm i} +1)^2 \right)^2$. As, in a given system, period and orbital separation scale as $P^2 \propto d^3$, this means that in systems with the same initial period, the primary star is, at the time when $P=P_{\rm min}$, squeezed into a smaller volume for smaller $q_{\rm i}$. This makes the mass transfer rate during the first phase of Case B mass transfer larger for smaller $q_{\rm i}$.

However, the maximum mass transfer rate, which is achieved after mass ratio reversal (cf. Sect. 4.1), is very similar in all three systems, even though the post-mass transfer periods (115 d, 87 d, and 72 d for secondary initial masses of 11$\,M_\odot$, 9$\,M_\odot$, and 8$\,M_\odot$, respectively) are not the same. Also the amount of mass which is transferred is independent of the secondary mass - it consists of the whole hydrogen-rich envelope of the primary.

Whether or not contact is reached is here mostly determined by the reaction of the secondary star to the accretion. Figure 9 shows that the primary with the lowest initial mass swells most during the mass transfer. The smallest initial mass ratio for which contact is avoided corresponds roughly to the condition that the mass accretion time scale of the secondary $M_{\rm 2}/\dot M_{\rm 2}$ remains smaller than its thermal time scale $G M_{\rm 2}^2 / (R_{\rm 2} L_{\rm 2})$. We designate contact due to this reason as q-contact.

As q-contact is established during the first phase of Case B mass transfer, it is quite distinct from delayed contact (Sect. 4.1). As q-contact evolves early during Case B mass transfer, it does not allow a major amount of mass to be accreted by the secondary. I.e., either the major part of the primary's hydrogen-rich envelope can be ejected from the system, or both stars merge. The situation that the secondary star can accrete significant amounts after a non-conservative early q-contact phase is unlikely since it would bring both stars even closer together.

In principle, there is also a maximum initial mass ratio for contact-free evolution of Case B systems: for $q_{\rm i}=1$, both stars age at the same rate and attempt to expand to red giants simultaneously. However, since the limiting value is very close to $q_{\rm i}=1$, we ignore this effect here as statistically insignificant.

Figure 8: Mass transfer rates as function of the transfered amount of mass for three Case B systems with 12$\,M_\odot$ primaries. The initial masses of the secondaries are 11$\,M_\odot$ (system No. 3), 9$\,M_\odot$ (No. 19), and 8$\,M_\odot$ (No. 27), as indicated. The initial period is 6 d for all three systems

Figure 9: Radii of the secondary stars as function of their mass for the same systems as shown in Fig. 8

Figure 10: Mass transfer rate as a function of transfered amount of mass, for five Case A systems with the same initial primary mass ( $12\,M_\odot$) and initial period (2.5 d). The initial secondary masses are 11.5$\,M_\odot$ (System No. 1), 10.5$\,M_\odot$ (No. 10), 9.5$\,M_\odot$ (No. 15), 8.5$\,M_\odot$ (No. 23), and 7.5$\,M_\odot$ (No. 31)

Figure 11: Radii of the secondary stars as function of their mass for the same systems shown in Fig. 10. The first maximum is achieved during rapid Case A mass transfer. The second maximum is achieved during slow Case A mass transfer due to the nuclear evolution of the secondary. The radius decrease after the second maximum is due to the Case AB mass transfer

4.2.2 Case A

The maximum mass transfer rate in Case A systems is achieved during the rapid mass transfer phase while the orbit is shrinking. Figure 10 shows the mass transfer rates of five Case A systems with identical initial conditions except for the initial secondary mass and thus the initial mass ratio $q_{\rm i}$. As shown in Sect. 4.2.1, the smaller $q_{\rm i}$ the larger is the maximum mass transfer rate since the primary is squeezed into a smaller volume. Also, more mass is transfered during the rapid Case A mass transfer for smaller $q_{\rm i}$. As for Case B, q-contact may occur for small $q_{\rm i}$ due to higher maximum mass transfer rates and due to larger thermal time scales of the secondary, the latter effect being the more important one. In Fig. 11, all systems shown avoid q-contact, but the relative radius increase during the rapid Case A mass transfer is clearly larger for the systems with smaller initial secondary masses.

The second expansion of the secondaries during the Case A mass transfer (Fig. 11) is due to their nuclear evolution during the slow mass transfer phase. It is ended by the onset of Case AB mass transfer. The secondary star in system No. 1 has already ended core hydrogen burning and expanded appreciably before the onset of Case AB mass transfer. The onset of Case AB mass transfer stops its expansion and prevents the system from evolving into premature contact. However, Figs. 10 and 11 demonstrate, that for too large mass ratios, massive Case A binaries evolve into premature contact as in the case of too small initial periods (cf. Sect. 4.1.2). I.e., while Case B systems evolve into contact only when their initial mass ratio is too large, Case A binaries can do so for too large or too small initial mass ratio.

Figure 10 shows also, that the total amount of mass which is transfered during case A and AB is nearly independent of $q_{\rm i}$. This can not be expected in general, as we shall see below that the primary mass after Case A and AB mass transfer depends on the core hydrogen abundance at the onset of mass transfer and on the initial mass ratio (cf. Fig. 14 below). While smaller initial mass ratios lead to smaller primary masses, the larger Roche lobe due to the lower secondary mass (for the same period) leads to mass transfer only later in the evolution of the primary and thus to larger post-mass transfer primary masses. Both effects cancel each other in the models shown in Fig. 10.