The Spin Period: Ups and Downs

The spin period modulation is the defining characteristic of an intermediate polar. Therefore an attempt to measure the spin period is a pre-requisite for accepting a CV as an IP.

Long-term tracking of the spin period is equally important, for several reasons.

At a pragmatic level, when we obtain, e.g., new X-ray data, we want to be able to phase it to a known spin ephemeris.
The relative stability of the spin period is an evidence for the white dwarf, as opposed to neutron star, nature of the compact object.
One can deduce the magnetic moment of the white dwarf if we assume that IPs are at or near their long-term spin equilibrium. This assumption needs to be confirmed, however. Patterson (1994) predicts that Pdot of IPs in spin equilibrium will not be steady.
Observational determiation of spin up/down gives a better handle on the magnetic moment, and hence allows an observational test of spin equilbrium (Patterson 1994). We might also be able to deduce something about the degree of departure from secular mean accretion rate in these systems.

Theory of spin evolution

For a recent review, see Chapter 10 ("Disrupted Discs: Stellar Spin Evolution") of Campbell 2018.

Warner & Wickramasinghe 1991 considered the interaction of braking and accretion torques to explain the distribution of objects on the Pspin-Porb diagram.

How to track spin period changes

Traditionally, spin period changes of IPs are tracked via O-C curves, which are then used to construct a new ephemeris (linear, quadratic, or cubic). However, Patterson et al. (2020) convincingly argue that it's better to track the spin period as a function of time directly.

Notes on Inidividual Systems

In the descriptions below, the "shelf life" of an ephemeris is indicated by the time it takes for the formal uncertainty to exceed 1 complete cycle. Note that this definition of a shelf life is rather generous. In practice, you should divide this by n - the value of n can be debeted, but 4 might be a reasonable choice.

Systems which have shown a complex spin history
- FO Aqr: Patterson et al (1998) found that both quadratic and cubic terms were necessary (the star was spinning down in the early to mid 1980s, then swithced to spin up in more recent years). Despite the long baseline, extrapolation of ephemeris is unlikely to be reliable, and in fact Williams (2003) was unable to establish a unique ephemeris through 2002. Patterson et al. (2020) found that it was once again spinning down during 2014 to 2017.
- EX Hya: Mauche et al. (2009) contains the most recent published spin ephemeris, according to which this system shows a secular spin-up, but a cubic term is needed for. a good fit.
Systems which have shown a secular spin-up
- BG CMi: A steady spin-up is seen over 15 years; see Hellier (1997a) for details. Further monitoring would be desirable to see if the Pdot is constant or not, since there is a slight difference between the value derived by Hellier (1997a) and that in Patterson & Thomas (1993). Patterson et al. (2020) showed that it exhibited steady spin-up through late 1990s, then stayed at a roughly constant period through 2010, then resumed spinning up (though not as rapidly as during the 1980s and 19902).
- DQ Her: Zhang et al. (1959) compiled the spin timings over 40 years and conclude that a cubic ephemeris is necessary. As of 1967, Pdot was -6.4x10^-13 but it will go to zero in about 70 years. Patterson et al. (2020) showed that it exhibited steady spin-up throughout the observed period, but the rate of spin-up varied from epoch to epoch. The steady spin-up may be related either to the elevated accretion rate following the 1934 nova eruption, or a slow contraction of the white dwarf following the heating during the nova eruption.
- GK Per: Secular spin-up was inferred for this source using a mixture of X-ray and U-band data by Patterson (1991), but with cycle count uncertainties. Recently, Mauche (2004) has made a secure determination of the spin-up rate using X-ray data alone. In the optical, U-band observations appear to be essential for a secure detection of the spin signal.
- AO Psc: Williams (2003) has confirmed the secular spin-up of this system, but was unable to provide a unique ephemeris. Patterson et al. (2020) showed that AO Psc exhibited steady spin-up throughout the observed period, but the rate of spin-up varied from epoch to epoch.
Systems which have shown a secular spin-down
- AE Aqr: Based on about 15 years of spin timing data, de Jager et al (1994) has derived a steady spin-down at the rate of Pdot(33s)=5.642(20)x10^-14 d d^-1. This is one important supporting evidence for the propeller model for this system. The ephemeris is very precise, but continued monitoring is highly desirable to confirm if the spin-down is really constant.
- PQ Gem: A steady spin-down is seen over ~10 years; see Evans, Hellier & Ramsay (2006) for details.
- V1223 Sgr: A secular spin-down was inferred by Jablonski & Steiner (1987) based on the sideband period ephemeris. The true spin period is reliably seen in X-rays but rarely? (ever?) seen in the optical. Patterson et al. (2020) showed that V1223 Sgr exhibited steady spin-down throughout the observed period, even though its luminosity is comparable to IPs that exhibt steady spin-up.
Systems for which spin up/down has not been measured
- HT Cam: Kemp et al (2002) have derived a very precise linear ephemeris, with a shelf life of over 30 years. They've established an upper limit on Pdot of 2x10^-11.
Systems for which a good ephemeris is not yet available
- XY Ari: The spin period was determined to be 206.3 +/- 0.1 s by Koyama et al (1991) using Ginga X-ray data. Subsequent X-ray and IR data only confirmed this value. Given the lack of an optical counterpart, it would be very difficult to do much better.
- V405 Aur: The spin ephemeris of Skillman (1996) has a nominal shelf life of 36 years; however, if we instead take the difference in period between this ephemeris and that of Allan et al (1996) as the true uncertainty, the shelf life drops to 10 years.
- V709 Cas: Kozhevnikov (2001) have observed this star during 1999 Oct 4-9 and detected the spin modulation at 312.77 +/- 0.04 s; the implied ephemeris has a 28.3 day shelf life.
- V1025 Cen: Buckley et al (1998) finds 2146.59 s spin period from photometry in 1995 and 1996.
- TV Col: I am not aware of any spin ephemeris for this system. The spin is primarily seen in X-rays, mostly with insufficient duration to establish an ephemeris. The only mention of positive detection of this period in the optical I've seen is in the "note added in proof" of Bonnet-Bidaud, Motch & Mouchet (1985).
- TX Col: Spin period inferred from X-ray and optical data (1984-1985) is 1911 +/- 2 s (Buckley & Tuohy (1989). Spin period is weakly seen in the optical, and various sidebands come and makes it difficult (if not impossible) to establish a long term ephemeris for this system.
- UU Col: Burwitz et al (1996) conclude that the probable spin period is 863.5 +/- 0.7 s from 4 nights of data during 1996 January 13-17. Shelf life of the implied ephemeris is 12 days.
- V2306 Cyg: The Norton et al (2002) ephemeris for half-spin period is based on 7 nights of V-band photometry at JKT, 2000 July 9-15. Shelf life of the implied ephemeris is 180 days.
- DO Dra: Table 6 of Haswell et al (1997) contains the most up-to-date table that I know of of the spin period, determined from X-ray and/or U band photometry (usually via its 1st harmonic). Data spanning many years are combined yet the uncertainty is such that the implied shelf life of the ephemeris is only 6 months. The lack of strong signal in white light photometry makes it difficult to establish a long-term ephemeris.
- V2400 Oph: Buckley et al (1995) and Buckley et al (1997) have identified 927.66 +/- 0.41 s spin period from polarimetry and 1003.299 +/- 0.003 s synodic period from photometry during 1991-1994. However, 1027 s period was also possible from these data. Assuming the 1003 s period, the implied ephemeris has a shelf life of 10 years.
- V1062 Tau: Remillard et al (1994) derived a period of 1.054+/-0.005 hr = 3794 +/- 18 s from 4 nights of I-band data, whereas Hellir, Berdmore & Mukai (2002) derived 3704 +/- 8 s from 2 days of (ASCA and RXTE) X-ray observations. More data are needed to establish the spin period.
- NY Lup: Haberl, Motch & Zickgraf (2002) observed this object from ESO on three consecutive nights, 1998 May 7-9. Shelf life of the implied ephemeris is 10 days.

Please send your comments, suggestions etc. to Koji.Mukai@nasa.gov and/or Koji.Mukai@umbc.edu

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