Introduction
It is always interesting – what other information can we exctract from the light curve and available data? How CG Draconis looks like? What happens there and what’s the deal with out of phase components and impossible orbital configuration? It turns out, the disc instability model provides us with a way to visualize this engimatic star.
Thanks to Jeremy Shears from the British Astronomy Association, I’ve got myself a copy of the “Cataclysmic Variable Stars” by Hellier (2001). This book provides fantastic encyclopaedic review of these stellar systems, including dwarf novae. It showcases an argument put forward by Osaki (1974) to support the disc instability model – back when researchers still weren’t sure if the outbursts are actually caused by changes in mass transfer rates from the secondary star or due to something going on in the accretion disc around the white dwarf.
Basically, the premise of the model lies in that the secondary star donates material which overfills its Roche lobe and then falls onto the accretion disc around the white dwarf, creating so called bright spot (sometimes called the hot spot.) The flow of the material is constant. Viscuous forces in the accretion disc limit the flow: if a critical mass is reached, the disc becomes unstable and more viscious. This increases the accretion rate on the white dwarf rapidly and produces the outburst. Once the increased flow is accreted and processed in the outburst, the system fades back into the quiescent state, material begins to build up in the disc again, and the cycle repeats.
Now, to illustrate that this model is actually preferrable to the secondary star instability hypothesis, Osaki offers us a quantitative check. Imagine the red dwarf secondary feeding material. The material builds up in the accretion disc, then it dumps onto the white dwarf, producing the outburst. We have two accretion flows here – on the disc, regulating the luminosity of the bright spot as:
Lspot ~ G * M / r * Q_disc, (1)
where Lspot is the luminosity of the bright spot, M is the mass of the white dwarf, r is the distance of the bright spot from the white dwarf, and Q_disc is the mass accumulation rate. And, then, the second flow when all this mass excess is dumped on the white dwarf:
Lmax = G * M * Q_wd / R, (2)
where Lmax is the luminosity of the system during the outburst, Q_wd is the accretion rate on the white dwarf and R is the radius of the white dwarf. Assuming that no material leaves the system, Osaki establishes the relationship between these flows, so that:
Q_wd / Q_disc ~ tau_R / tau_outburst (3)
where tau and tau_outburst are the period and the duration of the outburst respectively. Combining all the above equations, we get:
Lmax / Lspot ~ r / R * ( tau / tau_outburst ), (4)
…
Hence, in order to measure the distance of the bright spot from the white dwarf, r, the only missing component we need is the white dwarf’s radius. Webbink (1990) comes to help. He provides average mass of the primary, 0.91 +/- 0.08, measured for 21 dwarf novae with orbital period P > 2.4 h, which appears to be a rather tight range for our purpose. We can then convert the average mass of the white dwarf to the average radius, R, we can use, using the known white dwarf mass-radius relationship from Hellier (2001):
R = 0.779 ( M ^ ( 2 / 3 ) – M ^ ( 2/3 ) ) ^ ( 1 / 2 ) * 10 ^ 7 m (5)
Measuring CG Dra Bright Spot Distance
Measuring the shoulder of the bright spot, converting magnitudes to linear fluxes (obtaining both L), using the average R from above, and tau taken from the light curve, we can begin plugging some numbers… With the last observed U/N/A-type eclipse (sorry, this is my own nomenclature, means U-shaped, normal orbital hump and asymmetric profile, improvement suggestions are welcome) typically observed at quiescence, I get r = 2.77 x 10^9 cm, or about 0.04 solar radii. Not sure what helps to visualize this – this is about 4.36 Earth radii – that’s the distance of the point on the accretion disc from the white dwarf where the flow from the secondary star lands to.
What is great is that we can measure this length in the radii of the primary star – the white dwarf itself, so the unknown r cancels itself out. This gets us the distance from the bright spot to the white dwarf at 4.52 white dwarf radii at CG Dra quiescence. With this type of the eclipse the contribution of the bright spot is approximately 20% of the total light of CG Dra.
Repeating the same exercise for the Sharp-V/H/HA-type eclipse, also at quiescence, I get 5.43 white dwarf radii, and also about 20% contribution by the bright spot to the total light. If we search dwarf novae visualizations, this looks about right.
For the CG Dra outburst state, I took the V/L/S-type eclipse – V-shaped, low hump, symmetric. The contribution of the spot is only 5% now, as it is outshone by the accretion disc. Plugging the numbers, the bright spot is at the 9.11 white dwarf radii now. Assuming that the bright spot is located somewhere at the edge of the accretion disc, this means the disc grows about twice in size during the outburst. This is in line with the disc instability model, and also looks similar to how the accretion discs are known to behave in dwarf novae, e.g. EX Dra.
Next Steps
All the derived numbers are rough, as I have determined measurement points visually at different phases of the system, and there are myriads of assumptions, but should be a good starting point. I need to implement and standardize routines to determine bright spot ingress and peak points and averaging procedures. And, of course, all this completely ignores the engima of the secondary star in CG Dra and assumes we are dealing with a normal dwarf nova.
REFERENCES
Hellier, C. 2001, in Cataclysmic Variable Stars, Springer-Praxis
Osaki, Y. 1974, PASJ, 26, 429
Webbink, R. F. 1990, in Accretion-Powered Compact Binaries, ed. C. W. Mauche, Cambridge University Press