aa r X i v : . [ a s t r o - ph . S R ] F e b ACTA ASTRONOMICA
Vol. (2020) pp. 1–5 On the Periods and Nature of Superhumps
J. S m a k
N. Copernicus Astronomical Center, Polish Academy of Sciences,Bartycka 18, 00-716 Warsaw, Polande-mail: [email protected]
Received
ABSTRACTIt is commonly accepted that the periods of superhumps can be satisfactorily explained within amodel involving apsidal motion of the accretion disk provided the frequency of the apsidal motion inaddition to the dynamical term includes also the pressure effects. Using a larger sample of systemswith reliable mass ratios it is shown, however, that this view is not true and the model requires furthermodifications.
Key words: accretion, accretion disks – binaries: cataclysmic variables, stars: dwarf novae
1. Introduction
Superhumps are periodic light variations, with periods slightly longer than theorbital period, observed in dwarf novae during their superoubursts and in nova-likecataclysmic variables – the so-called permanent superhumpers.There are two, competing models for superhumps: the tidal-resonance modeland the irradiation modulated mass transfer model. According to the tidal-resonance(TR) model, first proposed by Whitehurst (1988) and Hirose and Osaki (1990),they are due to periodic enhancement of tidal stresses in an eccentric accretion diskundergoing apsidal motion (often incorrectly called "precession"). An importantingredient of this model is the 3:1 resonance between the orbital frequency of thebinary system and the orbital frequency of the outer parts of the disk which is essen-tial for the disk to become eccentric. This model, however, fails to explain manycrucial facts (cf. Smak 2017 and references therein) but in spite of that is com-monly accepted. The irradiation modulated mass transfer (IMMT) model (Smak2009,2017), based on purely observational evidence, explains superhumps as be-ing due to the periodically variable dissipation of the kinetic energy of the streamresulting from variations in the mass transfer rate which are produced by the mod-ulated irradiation of the secondary component.
A. A.
The periods of superhumps and their interpretation have been the subject ofnumerous investigations. According to the most recent ones (Murray 2000, Mont-gomery 2001, Pearson 2006, Smith et al.
2. The Data
The data set to be analyzed below consists of 26 cataclysmic variables, with in-dependently determined mass ratios, taken from the recent compilation by McAllis-ter et al. (2019, Tables 2 and B2) and supplemented from the compilation by Smith et al. (2007, Table 5). Included in our sample are also two helium CV’s: AM CVnwith q = . ± .
01 (Roelofs et al. q = . ± . et al. P orb < et al. P orb > et al
3. The Basic Equations and Relations
We begin by listing equations and relations which will be used in further anal-ysis. The superhump period P SH and the period of apsidal motion P aps are relatedby 1 P aps = P orb − P SH , (1)or, in terms of the corresponding frequencies ω = π / P , ω aps = ω orb − ω SH . (2)The superhump period excess defined as ε SH = P SH − P orb P orb , (3)is related to the apsidal frequency by ε SH = ω aps ω orb − ω aps . (4) ol. 70 et al. ω aps = ω dyn + ω press . (5)The ratio of the dynamical part of the apsidal frequency to the orbital frequencyas a function of the mass ratio and the effective radius of the disk is given by (Hiroseand Osaki 1990, Pearson 2006, Eqs.6 and 7) ω dyn ω orb = q ( + q ) / r / ∞ ∑ n = a n r ( n − ) , (6)where a n = ( n ) ( n + ) n ∏ m = (cid:18) m − m (cid:19) . (7)The effective radius of the disk is commonly assumed to be equal to the 3:1resonance radius which is given by r = / ( + q ) / . (8)
4. The Pressure Term ?
Following earlier papers (Murray 2000, Montgomery 2001, Pearson 2006, Smith et al. ∆ω between the observed apsidal fre-quency ω aps and the dynamical term ω dyn which, according to earlier authors, areexpected to represent the pressure term ω press .First, using the observed values of P orb , ε SH and q = M / M we determine theobserved apsidal frequency ω aps (Eq.4). Then, assuming – as is commonly done –that the effective radius of the disk is equal to the 3:1 resonance radius (Eq.8), wecalculate the dynamical contribution ω dyn (Eqs.6 and 7) and, finally, the difference: ∆ω = ω aps − ω dyn .Results, presented in Fig.1, can be summarized as folows:(1) For systems with P orb < ∆ω show large scatter and/orappear to be correlated with the mass ratio.(2) Systems with P orb > ∆ω = − . ± . ∆ω = − . ± . (4) The theoretical ω press = f ( q ) relations (Montgomery 2001, Pearson (2006)fail to represent the real data. replacing r with r tid for systems with orbital periods above the period gap changes the resultsonly slightly. the problem with the AM CVn systems was already noted by Pearson (2007). A. A.
Fig. 1. The residuals ∆ω (see text for details) are plotted against the mass ratio. Red symbolsrepresent the two helium CV’s. Red and green lines are theoretical ω press = f ( q ) relations from,respectively, Montgomery (2001) and Pearson (2006).
5. Discussion
Until now, as mentioned in the Introduction, it has been believed that the pe-riods of superhumps can be satisfactorily explained when the apsidal frequency isassumed to be the sum of the dynamical and pressure terms. Results presentedabove imply that this is not true. Therefore the basic model involving the apsidalmotion of the disk requires substantial modifications.
Fig. 2. The effective radii r e f f (see text for details) are plotted against the mass ratio. Red symbolsrepresent the two helium CV’s.
Searching for possible clues we follow Pearson (2006) and determine the effec-tive radii r e f f at which the apsidal frequency calculated using only the dynamicalterm (Eqs.6 and 7) would be equal to the observed frequency. Results, presented in ol. 70 all values of r e f f , including those represent-ing the two AM CVn systems, fall between ≈ . ≈ .
4. The significance ofthis results is, however, not yet clear.REFERENCES
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