Support for Magnetic Complexity in Cataclysmic Variables

Cataclysmic variables (CVs) are binary systems consisting of a white dwarf and a secondary "normal" star. Material from the secondary star gets stripped away and forms an accretion disk around the white dwarf (see Figure 1). Astronomers observe variations in brightness of the system due to activity in the accretion disk. 

Figure 1. Typical Cataclysmic Variable system. Credit NASA

One observable characteristic of a CV is its orbital period, the amount of time it takes one object to go around the other. An interesting fact to come out of studying CVs over many decades is that there is a paucity of orbital periods between roughly 2 and 3 hours (see Figure 2). This so-called period gap has been a mystery to astronomers for since the late 1970s.

Figure 2. Histogram illustrating the period gap between roughly 2 and 3 hours Credit Carraffo et al. 2018

There are some ideas about why such a period gap may exist. The prevailing hypothesis is centered on the notion of loss of angular momentum. The standard explanation is that at less than 2 hours the loss of angular momentum is dominated by gravitational radiation while at greater than 3 hours it is dominated by magnetic braking. Let's pause for a moment to consider these terms. 

Gravitational radiation (a.k.a., gravitational waves) is a byproduct of Einstein's theory of general relativity (GR). GR views gravity not as a force, but instead a result of the curvature of space created by objects with mass. When you have two massive bodies orbiting each other rapidly (i.e., a CV system), they create ripples in space. Such ripples were detected for the first time in 2015 to great fanfare. A result of gravitational radiation is that energy, and thus angular momentum, is lost from the binary system.

Magnetic braking is another way for stars to lose angular momentum.  Stars are spinning balls of plasma (ionized gas) that give rise to magnetic fields around themselves. The magnetic field lines can strip away ions in the star's atmosphere and carry them away. Again, this mass loss leads to a loss in angular momentum. This phenomena can be greatly exacerbated by stars with strong magnetic fields.

Back to CVs, remember that astronomers think that gravitational radiation dominates systems with periods less than 2 hours and magnetic braking is responsible for those above 3 hours. So why the period gap? The proposed reason for the gap is that around a period of 3 hours the magnetic breaking "turns off", the secondary star contracts and its orbit shrinks resulting in a quicker (< 2 hour) period. However, there is a dearth of evidence supporting this idea of an interruption of the magnetic field at about 3 hours. An alternative explanation offered by Taam & Spruit (1989) suggests that as the secondary star's rotation decreases, its magnetic complexity increases which leads to less effective loss of angular momentum. In other words, the magnetic field lines get gnarly and cannot shuttle the mass away in an orderly fashion.

In a recent study, Garraffo et al (2018). studied the CV period gap by attempting to model this magnetic behavior using Zeeman-Doppler imaging (ZDI). This also deserves some explanation. The Zeeman effect is the observation that individual spectral lines can be split into multiple parts in the presence of a strong magnetic field. When this effect is observed for a system over time, it can be used to essential re-create the structure of the underlying magnetic field (see Figure 3).

  

Figure 3. Zeeman-Doppler imaging reconstruction of a young star's magnetic field. Credit Wikipedia

In the models that Garraffo et al. (2018) created, they showed that surface magnetic complexity did indeed increase along with the decrease in orbital period. These results support the idea that magnetic complexity can suppress mass loss. Consequently, this reduction in the loss of angular momentum could result in the cessation of accretion to the primary. Recall that the accretion disk is how astronomers observe CVs, so when it is "off" no observations could be made. This is a very plausible explanation for the existence of the period gap. As the system continues to lose angular momentum, eventually the components are in close enough proximity for gravitational radiation to take over. As always, more studies will be needed to confirm if magnetic complexity is indeed responsible for the infamous CV period gap.

One Disk to Rule Them All

Figure 1. A typical dwarf nova. Credit NASA via www.skyandtelescope.com

In a study accepted for publication in the Monthly Notices of the Royal Astronomical Society (MNRAS), astronomers Aranzana et al. reported on Fourier time lags in the dwarf nova SS Cyg.

SS Cyg is a well studied variable star system in the constellation Cygnus the swan. Specifically, it is a special type of cataclysmic variable called dwarf novae (DNe). DNe consist of a binary system made up of a primary white dwarf that every so often siphons off material from its low-mass companion that has overfilled its Roche Lobe. This material forms an accretion disk that swirls around the primary star. It is this accretion disk that will occasionally brighten across a variety of wavelengths. These "brightenings" are associated with outbursts and, along with DNe, are observed across a variety of astrophysical phenomena such as X-ray binaries (XRBs) and Active Galactic Nuclei (AGN).  Although these are widely different phenomena, astronomers have evidence to support the idea that they share the same underlying accretion disk that gives rise to the outbursts. Subsequently, these are interesting targets for research since studying any of these objects could shed light on all the others.

There have been several studies of that give credence to the notion of an accretion disk connection between CVs, XRBs, and AGN. For example, all three objects have demonstrated that increases in brightness are correlated with increases in variability. This suggests mass transfer to the accretion disk resulting in the aforementioned increases in both brightness and variability. Furthermore, fluctuations in the outer, cooler parts of the disk are hypothesized to travel inwards towards hotter regions. This leads to emission of soft photons (lower energy) from the cooler areas of the disk before hard photons (higher energy) from the hotter, inner regions of the disk. Subsequently, there have been X-ray studies where soft photons arrive at astronomers' detectors slightly before hard photons, a so-called 'hard/positive' time lag. Because this phenomena presumably requires mass transfer, it is only observed during an outburst.

However, what about periods of no mass transfer? There have been reports of 'soft/negative' time lags in CVs, meaning that the hard photons arrive before the soft ones. There is debate over the cause of this soft lag. One explanation is that emission from the source (white dwarf) lights up the disk, which in turn creates a reflection spectrum. However, this idea suffers from the report that such a spectrum would lie outside the binary orbit. An alternative explanation is that the photons from the white dwarf are thermally reprocessed in the disk, leading to hard photons being emitted from the accretion disk before soft ones (i.e., a soft/negative lag, see Figure 2).

Figure 2. Diagram showing emission of photons from the white dwarf being reprocessed in the disk. Since the higher energy/hard (u) photons are closer to the source, they get reprocessed before the lower energy/soft (r) photons resulting in the 'soft lag'. Credit Aranzana et al. 2018

The purpose of the study was to gather more evidence that could improve upon or rule out either of the models. SS Cyg was chosen as an optical target because it is one of the brightest DNe during quiescence (no activity). Researchers used the 4.2 meter William Herschel Telescope located in the Canary Islands, Spain over the course of two nights to take images using SDSS filters Ultraviolet (u), Green (g), and Red (r). In astronomy, color is typically represented by combining two filters. For example, r & u is the combination of the red and ultraviolet filters. Figure 3 shows the results for the three different color combinations averaged over both nights. At a frequency of 4 x 10^-3 Hz there is about a -6 second time lag in the g & u combination and -4 second time lag in the r & u combination.

Figure 3. SS Cyg during quiescence demonstrating a significant -4 second time lag for r&u and -6 second time lag for g&u at 0.004 Hz. The r&g Credit Aranzana et al. 2018

Due to the associated error bars, the r & g combination is consistent with no lag. The researchers suggest that this could be due to the fact that the source is also emitting primarily in this band. However, the overall results confirm the existence of soft time lags in the dwarf nova SS Cyg. Further lines of inquiry include expanding the number of CVs studied to see if the time lag is repeatable and consistent and also continued monitoring of the relationship with XRB and AGN time lags. Continued study of these time lags in various types of astronomical phenomena with accretion disks could help understand the underlying physics that unite them.

Binning

Binning groups pixels together which has the effect of making the group appear as a bigger single pixel as shown in figure 1 below (Templeton & Beck 2014).  Binning is done to decrease the resolution: 

So by increasing the pixel size while keeping the focal length constant, the resolution goes down.  This might seem backwards, but lower numbers imply higher resolution since less of the sky is "landing" on each pixel creating a sharper image.  The benefit of binning is that you gain better sensitivity so you can do shorter exposures (Templeton & Beck 2014).  The drawback is that you do lose some resolution so you have to be careful not to blend close by stars together if you are trying to do accurate photometry (Templeton & Beck 2014).  Another important thing to remember is that you have to redo all of your calibration frames and test the linearity for each set of bins (Templeton & Beck 2014).  In other words, a dark frame at 1x1 binning is not equivalent to a dark frame at 2x2 binning.

Figure 1.  Examples of binning (Templeton & Beck 2014)

References

Templeton, M. & Beck, S. (2014), The CCD Guide to Photometry Version 1.1, Cambridge, MA

Linearity testing

Figure 1.  CCD bucket analogy (Howell 2006)

Each pixel of a CCD sensor is like a bucket, but instead of collecting rain water, these buckets collect electrons that are generated by the incoming photons.  Just like a bucket can only hold a finite volume of water, a pixel can only collect a finite number of electrons before it "overflows."  When the bucket overflows, the measurement of  the electrons is no longer linear.  What does that mean?  Consider in the water example, a certain number of rain drops will produce a certain change in volume of the amount of water in the bucket.   This relationship holds up to the point where the bucket overflows and the incoming rain drops cannot be measured because all of the space has been taken up.  The same is true for pixels in the CCD sensor:  There is a linear response between the amount of incoming photons and the measured number of electrons per pixel up to a certain point where the relationship breaks down and the pixel "overflows" or "blooms" (see Figure 2 below).

Figure 2.  Example of blooming (AOweb)

Some types of CCD sensors have what is called an Anti-Blooming Gate (ABG) to try to minimize the effects of blooming.  However, since photometry is concerned with precise counting of the number of photons, it is better, but not absolutely necessary, to have a Non Anti Blooming Gate (NABG) to get an unadulterated measurement.  The manufacturer's CCD specifications will include the Full Well Depth of the sensor, or the expected number of electrons that can be collected before the pixels bloom.  In either case (ABG or NABG), it is best practice to actually test the camera to find out where the pixels saturate and if there is any non-linear behavior before the point of saturation.  This will enable better photometry since the "overflow" point is well established and can be avoided.

In practice, most photometry software packages do not actually output electrons, but instead use Analog to Digital Units (ADU).  ADU is simply a numerical representation of the voltage created by the electrons that accumulated in the pixel.  The ADU can be converted to electrons by multiplying by the CCD's gain, another specification given by the manufacturer.  In the case of TECMO, the SBIG ST-8XE (NABG) has a specified Full Well Depth of 100,000 electrons and a gain of 2.5 electrons per ADU.  Therefore, the pixels should bloom at approximately 100,000/2.5 = 40,000 ADU.  To test this, the telescope and CCD were pointed at a diffuse, uniform light source.  For TECMO, white sheets of paper were used to diffuse the light coming from a light bulb.  A good starting place is when a 10 second exposure yields an average ADU measurement of about 10,000.  In this example, the "brightness" was reduced by adding more sheets of paper over the telescope's aperture.  After this, the exposure time was doubled and the ADU measurement was taken again.  What was expected?  Since the exposure time was doubled, the amount of light reaching the pixels was doubled, so the initial ADU value should double as well to around 20,000.  Now another 10 seconds is added so that the exposure is 30 seconds and the ADU measurement should roughly triple.  Keep adding 10 seconds until this relationship breaks down and the ADU measurement no longer goes up because the pixels have "overflowed".  The figure below shows the results of the TECMO linearity test of its SBIG ST-8XE.

The sensor clearly saturates at just over 60,000 ADU.  However, after fitting a straight line to the data, there also appears to be some non-linear behavior before the point of saturation.  It is not entirely clear from this data exactly where the linearity ends, but a reasonable estimate is about 50,000 ADU.  This is a "deeper" full well depth than reported by the CCD manufacturer (SBIG).  However, it is probably best that their estimate is conservative.  In fact, it is smart to use the more conservative number to calculate exposure times for the targets of interest.  For example, if the goal was to expose a target until it reaches half of its full well depth (a typical practice in photometry), it would be wise to use 40,000/2 = 20,000 ADU.  When in the business of counting photons, it is always safer to err on the side of under-exposure in order to maintain the all important linear relationship between the incoming light and measured electrons.

References

1. AOweb:  http://astronomyonline.org/astrophotography/ccd.asp, accessed 11 November 2014
2. Howell, S. B. 2006, Handbook of CCD Astronomy, Cambridge University Press, UK