White Dwarf Story

B. Gary, 2024.05.23

 This web page is structured to motivate celestial mechanic modelers to explain observations of white dwarfs with transiting dust clouds. It assumes only a modest familiarity with the WD observations that need to be explained. It is assumed that dust clouds are produced from collisions and that interloper planetesimals in highly eccentric orbits are involved. An ever-present "Roche debris disk" is assumed to exist, created and maintained by millions of years worth of planetesimal tidal disruptions at periastron passage. No attention is given to alternative mechanisms for dust production, and I also neglect the mysterious matter of how a dust cloud can be maintained for months without spreading out into nothingness on daily timescales due to Keplerian shear. 

Step 1: Natal Kicks 
WD With Low Packing Fraction (J0139) 
WD With Higher Packing Fraction (WD1145) 
WD With Even Higher Packing Fractions (J0328)
Scenario Leading to Interloper/Debris Disk Interactions  
Questions for Modelers
Deacription of Packing Fraction  
References 

1. Natal Kick

Imagine a WD that is orbited by millions of planetesimals that used to be in far-out orbits (before the planetary nebula transition) but are now in highly eccentric ones with millions having their periastron within one A.U.. At any randomly chosen time one of the planetesimals orbits so close to the WD that it feels tidally disrupting forces during periastron passage. Let's refer to it as an "interloper."

Imagine also that previous interlopers have left debris that is at the tidal disruption distance in orbits that have been circularized (due to Poynting-Roberston drag and the Yarkovsky effect). Encounters with this debris shortens the interloper's orbital period., so encounters become more frequent.

Every periastron passage is brief, lasting less than a minute. But the distance covered is long due to its high speed (440 km/sec). The interloper's Hill sphere size becomes irrelevant in determining whether debris undergoes an orbit change; only impacts matter. Most impacts are grazing because "impact parameter" values are random.

Whenever a periastron passage produces one or more collisions this may initiate a cascade of collisions. Every collision produces a dust cloud. If the debris has a high "packing fraction" then many collisions can occur. If the "packing fraction" is low, then there may be periastron passages when no collision cascades occur. (For my description of "packing fraction" go to link below.)


2. WD With Low Packing Fraction (J0139)

Consider the light curve for a WD with a low "packing fraction" (J0139).  Every 107 days, approximately, there are dozens of dips, with each dip presumably the result of a collision. The series of dips (the individual ones aren't shown in this graph) last about a week. Notice that the 107-day timing is inexact (e.g., #2 is late). This is because some collision cascades take longer to reach a peak than other cascades. Also, notice that sometimes no cascade occurs (e.g., #6).  The greatest deviation from a repeating 107-day occurrence is #9. (It's possible that the early set of dips was produced by a second interloper.)


Figure 1. Date-averaged magnitude for 2.5 observing seasons. Each group of dips (fainter than an "out-of-transit" baseline) consists of dozens of individual dips. The most-likely interval between dip groups is 107 days, but real departures of a few days exist. The duration of each dip group is about a week.

The next graph is a detailed look at one of the dip groups. The smoothed trace (blue) has a FWHM = 6 days, which is typical for the other dip groups. Every night's observing session included many dips. A periodicity for the individual dips has not been determined.


Figure 2. An 8-day set of observations of J0139 shows that every night has many dips. The smoothed (blue) trace has a FWHM = 6 days.

Th next graph is a light curve for one of the observing sessions in the previous graph. There is evidence for 6 dips, with plenty of overlap.


Figure 3. Light curve for one night's observing session, showing 6 dips with AHS function fits (having plenty of overlap). 


3. WD With Higher Packing Fraction (WD1145)

WD1145+017  apparently has more fragments per unit of radial distance from the WD than J0139 (i.e., a greater "packing fraction")._This is revealed by two observational facts: 1) during a dip group the fraction of time for OOT is greater (typically 30 to 40 %), and 2) each periastron passage (at 124-day intervals for WD1145) has an activity function that is a substantial fraction of the periastron interval (FWHM = 60 days for WD1145 instead of 6 days for J0139). The following graph is essentially chosen at random from an active year. Approximately 1/3 of the time is OOT. 



Figure 4. A night's LC from an active year for WD1145. OOT is present ~ 1/3 of the time.


The next graph shows "dip activity level" versus date for 4 observing seasons. At the bottom is an offset sine wave with a 124-day period. It does a good job of predicting dip activity. The black trace is the sum of many Gaussian functions, spaced at ~ 124-day intervals, with amplitude adjusted to fit measurements. Small adjustments to the best overall repeat interval of 124 days are permitted because collision cascades are expected to have variable timescales for reaching peak activity. Differing amplitudes is also allowed because each collision cascade can produce different peak levels due to the randomness of how fragment orbits are packed. Each Gaussian function is also allowed to have a slightly different width. A typical FWHM width = 60 days.



Figure 5. Dip activity for the first 4 season's of ground-based measurements. The sinusoidal trace (tan, at bottom) has a period of 124 days), and it is correlated with measurements of dip activity. The black trace that fits data was produced by a repeating pattern of Gaussian shapes with height and widths adjustments. Small date shifts were also applied to improve fit. 

To see details of the above graph, one observing season at a time,  go to this link.

If the 124-day periodicity of dip activity is real then it should show up in a periodogram. It does!

Figure 6a. Lomb-Scargle periodogram of the WD1145 dip activity plot for the first 6 observing seasons of ground-based data in my archive. The main feature is a period of 0.346 years, or 126 days. The presence of features with longer periods are due to the year-and-longer timescale variations (apparent in Fig. 5).


The next graph is a periodogram for a version of dip activity in which each season's level was either increased or decreased in order to achieve approximate equality of peak activity level for that season. The purpose in doing this is to eliminate periodogram features produced by year-and-longer timescales. This should produce a "cleaner" solution for the main feature.




Figure 6b. Lomb-Scargle periodogram of the WD1145 dip activity plot for the first 6 observing seasons of ground-based data in my archive after normalizing individual seasonal variations to the same approximate activity level. The main feature is a period of 0.342 years, or 125 days. The presence of features with longer periods are reduced, as expected, which allows for a "cleaner" solution for the main periodicity.


Indeed, the expected "cleaning" of the periodogram was achieved, and the main period stands out more prominently. The interloper period is now estimated to be 125 days. (The second highest feature has P = 94 days; if real, iy could be a second interloper.) 

The next graph shows WD1145 dip activity level for 11-years, using a log scale for representing activity.  The green trace is an eyeball fit using a Poisson function.



Figure 7. Activity for the last 11 years, using a log scale for activity. The thick green trace is a Poisson function (lamda = 2), added to an ever-present 0.25 % activity level.


The above graph's green trace calls to mind a scenario in which a big collision occurred in 2015, and that it initiated a sequence of secondary collisions. Yes, but the modulation of the green trace, i.e., the black trace, inspires a more nuanced model: a big collision happened in 2015 by the periastron passage of an interloper, and every subsequent periastron passage instigated a collision cascade (of duration ~ 60 days, FWHM) whose dip activity level exhibited a Poisson amplitude (green trace) versus date, whose shape was influenced by an exhaustion of fragments in the WD's "Roche debris disk" periastron  vicinity (described below).

During the peak activity years of 2016 and 2017 it was common to observe a dozen or more dips per 4.5-hour orbit. Since dips lasted for a couple weeks, typically, it was tempting to imagine that the number of fragments that were capable of producing dust clouds numbered 1000 or more. If after a few years there was an "exhaustion" of fragments that could be involved in a interloper-initiated collision cascade it may be possible to model how many fragments (capable of producing dust clouds) were present prior to the 2015 collision.

Support for a big collision in 2015 comes from a waterfall plot that shows a set of drift lines in "orbit phase/date space" that radiate from the date 2015 August. A reviewer of our article that reported the first comprehensive set of ground-based LC observations of WD1145 (Rappaport et al., 2016) was the first to notice this pattern.


Figure 8. Waterfall plot of dip orbit'phase location versus date, allowing a pattern of diverging drift lines to be displayed. Each drift line is for a dip that lasted many days. A drift line's slope is determined by the dip's orbital period departure from the waterfall's nominal period.

Many cases of a sudden appearance of branching drift lines were observed during the years of high dip activity. The next figure is one example.


Figure 9. A collision must have occurred on Day # 416 that produced 4 fragments that produced dust clouds that lasted ~ 10 days.

The slope of each waterfall plot drift line corresponds to an orbit period. During the 2015/16 observing season we measured many such periods. Here's a plot of how the periods that are associated with the A-disk varied with date:


Figure 10. Detail of relationship between periods of A-system dust clouds and the K2 A period (vertical red line) for the 2015/16 observing season.
    

We never argued for the reality of a slowly decreasing A-disk period, but if real it might help with modeling. For the past 10 years almost all observed dips have a period similar to the ones in the above figure. We occasionally observe dips with longer periods. For example, the next graph shows all 6 Kepler-based periodicities (converted to radial distance from the WD):


Figure 11. The 6 Kepler periods correspond to these distances from the WD shown here (for an assumed WD mass and radius). The symbols are based on measured drift line slopes (orbit periods).

What could possibly produce this distribution of dip activity versus orbit period? Lets start with asking what the "Roche debris disk" might look like after a long interval without interloper passages during which debris from earlier passages had sufficient time to circularize.


Figure 12. The dashed trace is a hypothetical plot of number density for debris after a long interval of no interloper periastron passages, which is long enough for the debris shed by  previous interlopers to have circularized. The solid trace is a hypothetical plot of debris number density after a new interloper has made a few periastron passages (at ~ 97.9 x WD_radius). A "clearing" of debris due to collisions is shown at the periastron location. 

In the above figure it is assumed that prior to this interloper's appearance at a periastron distance of 97.9 WD radii a Roche debris disk had already formed from prior interloper passages. It is assumed that this prior shedding of debris had sufficient time to circularize, leading to the dashed number density trace. The new interloper collided with debris at its periastron location, creating a "cleaning out" gap in the number density plot. We can presume that this process involved collision cascades that spread out in both directions: inward and outward.

Now let's consider a new scenario. Let's place the new interloper's first periastron passage to be at ~ 103 WD radii. It will "clear out" debris at that location. Eventually, during its outer orbit it loses speed from an encounter, and this causes it s next periastron passage to be at 102.3 WD radii. It then proceeds to "clear out" debris at that location. After four more similar losses of angular momentum in its outer orbit it begins a series of periastron passages at 97.9 WD radii. This is portrayed in the next graph.
 
Figure 13. Hypothetical distribution of debris in relation to the Kepler ABCDEF rings. The upper curve is a suggestion of how debris might be distributed from interloper sheddings and the eventual circularization of this debris if interlopers didn't collide with the circularized debris. The lower trace shows how this distribution could be affected by interloper collisions with debris during intervals of periastron passages at locations just outside the Kepler period locations. The down arrow is my suggestion for the interloper's current periastron location.

The above graph shows a hypothetical distribution of the "Roche debris disk" number density (which will of course also have a size distribution). During each periastron passage the interloper will spend more time (~ 10 minutes) at its closest approach distance, which is assumed here to be just outside the A-disk, than it spends in the proximity of the outer Kepler disks. If it has already cleaned out debris orbiting at its present periastron location there won't be any debris to collide with at that location. However, if there is a slow secular inward migration of periastron distance the interloper will eventually collide with debris at slightly unward locations (i.e., at the outer edge of the A-disk). If one of these collisions starts a collision cascade it can only progress inward because that's the only direction where debris is present.

The interloper spends much less time passing through the outer debris disks (DBCEF). It therefore has fewer collisions with debris at these locations. When Kepler observed there was a low level of collisions with these disks, but after a collision in early 2015, and a bigger one in August of 2015, the A-system collisions overwhelmed ground-based observations and it became difficult to detect the still small level of thee out orbit dust clouds. At the present time, when A-system dust cloud dips are negligible, the outer orbit dips remain difficult to detect because only small aperture amateur telescope are used for monitoring wd1145.

 The next graph provides evidence for the previously described scenario.


Figure 14. Waterfall plot for the first season of intensive ground-based observations of WD1145. During 2015 November to 2016 January, most dips had a period similar to the Kepler A period. and they all originated at the same phase location. Afterward, all periods were shorter (left-leaning slopes) and there were no more origins of dust cloud dips at the original phase location.

Ground-based measurements from 2015 November until 2016 mid-January showed that most dips originated at the same Kepler A orbit location and remained in orbits with this period. A few dips drifted leftward in the waterfall plot, indicating that they were in smaller orbits. In other words, collision cascades began at the outer edge of the A-disk, where the where the interloper had recently made a periastron passage (in August?), and the collision cascade is with debris in slightly smaller orbits. By mid-January there were no more collisions at the previously active collision location, and in its place is a group of persistent collisions at locations withing the A-system disk. This is what the previous scenario predicts should happen.

WD1145 is the most-observed and most-studied WD with transiting dust clouds. It is referred to as the prototype for such systems. It is therefore a reference system for describing other WDs with transiting dust clouds - one of which will be briefly described next.


4. WD With an Even Higher Packing Fraction (J0328)


J0328-1219 is a WD with a presumed even higher packing fraction. Dips are present almost all the time; OOT levels are sometimes not present. The next figure is a LC illustrating this.


Figure 15. A night's LC of WD J0328, showing an almost absence of OOT time during the 9-hour observing session.


An analysis of this WD's "dip activity level" versus date has not been performed, so it is not yet known if an interloper is present. If an interloper is present it is producing dip groups with long duration (comparable to, or longer than, the interloper's orbit period).


J0107+2107 is another WD with a high packing fraction. It has so many closely-spaced dips that OOT time has maybe never been observed. I won't present a LC for it here, but it's worth mentioning because it illustrates the concept that WDs with transiting dust clouds extend to those with extremely dense dip activity, which means that very high packing fractions are possible.


5. Scenario Leading to Interloper/Debris Disk Interactions


This web page's introductory section suggested that a sequence of events leading to interloper interactions with a preexisting "Roche debris disk" starts with a "natal kick" to the WD when it undergoes an anisotropic shedding of (half) its atmosphere, producing a planetary nebula; this was described by Akiba, McIntyre and Madigan (2024). WD velocity changes ("kicks") within the range of 1 to 3 km/s were capable of changing the orbits of distant planetesimals enough so that some of them befan to orbit retrograde. This means that there must be a population of planetesimals that had orbits bringing them close to the WD's Roche radius (distance where tidal disruption could occur).

We can now speculate what the second step might be. The first time a planetesimal passes close to the WD's Roche radius it may shed some of its material. This material will initially be in the same highly elliptical orbit as the parent planetesimal. However, after many periastron passages (of both the parent planetesimal and shed debris) the debris, because it is smaller and less massive, will be affected by Pointing-Robertson drag and also the Yarkosky effect. Neither of these effects will change the orbit of the planetesimal because it is too massive. But PR drag will reduce the orbital momentum of debris that is smaller than some size, such as 1 km radius (?). The Yarkovsky effect will reduce orbital momentum of fragments rotating in a retrograde direction, and that are smaller than some size (possibly also 1 km in radius?). After many periastron passages the debris, smaller than some size, will "circularize" their orbits to a radius close to the WD's Roche radius. This may require millions of years.

During that millions of years that small debris circularizes, there mau also be many additional interloper tidal passages and shedding events. A steady-state of interloper sheddings, and simultaneous circularizations, must exist.

The third step is about an interloper passing through the disk of smaller-sized debris at the WD Roche radius. That's what i've been referring to as the "Roche debris disk." Each planetesimal periastron passage is an opportunity for impacts. When the larger planetesimal impacts a smaller fragment, the smaller fragment will be smashed into to a set of smaller fragments with a "particle size distribution," or PSD, that may extend down to sub-micron radii. The smallest particles can be thought of as a dust cloud which can both scatter and absorb the WD's light (producing dips), whereas the larger ones, with radii of meters and tens of meters, can be viewed as an addition to the Roche debris disk. The large debris will be in orbits whose parameters are changed by random amounts from the parent's orbit. The new debris may be in orbits that cause them to impact other debris. If so, then a cascade of collisions could begin. Whether or not a collision cascade is initiated by the initial collision (the planetesimal and one debris fragment) depends on the debris disk's "packing fraction." For a high packing fraction it is more likely for a collision cascade to occur. 

The circularized fragments that constitute the "Roche debris disk" may be considered the product of a steady-state production process (with timescales of millions of years) and a steady-state destruction of these fragments due to impacts with more recent interlopers. The rates of production and destruction will be one factor determining "packing fraction."

Here's a diagram of the periastron path of an interloper in an elliptical orbit traversing the WD1145 "Roche debris disk." The 6 Kepler disk systems may simply be the inner part of WD's "Roche debris disk."

Figure 16. A possible path of a WD1145 interloper, which assumes it passes through the long-lived "Roche debris disk" at the WD's Roche radius. I assume that the 6 Kepler periodicities correspond to the inner part of the "Roche debris disk" and that the disk will last for much of the WD's lifetime. (A more realistic diagram would include several interlopers in elliptical orbits, with farther-out periastron distances and random longitude orientations.)

Over time each interloper will migrate inward, both slowly and abruptly, due to encounters with material in it out orbit portions, and it will "take its turn" being tidally disrupted and replenishing the "Roche debris disk." In fact, there may be times when more than one interloper is passing through the "Roche debris disk" (at different periastron passage intervals). At any given time there will be gaps in the Roche disk due to passes by previous interlopers. The slow inward migration of the interloper's periastron distance will produce collision cascades that only progress inward. This speculation is supported by WD1145 observations. Since debris in this disk may have taken millions of years to circularize after being tidally separated from an interloper, the "Roche debris disk" may consist of fragments from many tidally disrupted interlopers.


5. Questions for Modelers

The previous section has many loose end questions that will be a challenge to modelers. For example:

1) How often will the "natal kick" produce planetesimals that come close to the WD's Roche radius?
2) How long will these planetesimals (which I rename "interlopers") stay close to the Roche radius?
3) What sizes of debris will be affected by PR drag and the and Yarkovsky effect?
4) How long will it take for PR and Yarkovsky to circularize debris?
5) How much debris will be shed by an interloper during each periastron passage, and how does that limit the "lifetime" of the interloper?
6) How big a radius span will the "Roche debris disk" have?
7) Will gaps form in the debris disk, leading to a set of narrower debris disks (resembling WD1145's set of 6 lesser disks)?
8) What will be the PSD from the debris after a interloper/debris collision?
9) How often can we expect the initial collision (interloper/debris) to produce secondary collisions (i.e., a collision cascade)?
10) Is it reasonable for debris/debris collisions to form dust clouds that match observed dips?
11) Can packing fraction, or the PSD function, be inferred from the duration of dips associated with a single periastron passage (e.g., 6 days vs. 60 days)?
12) Can packing fraction, or the PSD function, be inferred from the variability of timing for peak dip activity?

Then there is another set of questions (not suitable for celestial mechanics analysis) that have puzzled modelers for the last decade, such as:

1) What is the mechanism for dust production (collisions, or sublimation and recondensation after thermal escape)?
2) What is the dust destruction mechanism (sublimation, Keplerian shear, PR, Yarkovsky)?
3) Why is there an absence of small dust (radii < 1 micron for WD1145, radii < 0.1 micron for J0328)?


Tom Kaye and I are observationalists. We want to create puzzles for modelers. I think we and other observers of "transiting WD dust clouds" have created a big enough puzzle for modelers to "take on" the task of answering some of the above questions.


6. Description of Packing Fraction

Imagine a 2-D disk with circular radii from a1 to a2. Within this 2-D space it is possible to have orbits described by semi-major axis distance, eccentricity and orientation of periastron (line of apsides). Each orbit has a location in 3-D parameter space. If we assign a mass to the orbiting object then we have also specified a Hill sphere radius. No other object is allowed within this 4-D parameter volume. (Actually, there's one more parameter which we haven't specified: the object;s location in the specified orbit; but let's keep things simple and neglect that parameter).

When a second object is created it can't be placed in the same 4-D volume without risking eventual collision with the first object. By placing the second object in a different (non-conflicting) orbit we've used up another increment of 4-D space that was initially available.

I think the rest of this argument is obvious. There is a maximum number of orbits that can exist without collisions (assuming each object has a specified mass). The "packing fraction" can be said to be 1.0, or "full." when the maximum number of objects are present. Clearly, if we start with a packing factor of 1.0, any change in orbit of an object will lead to at least one collision. And that collision could lead to a third collision, etc..

It is also obvious that with only two objects located at random locations in the disk, when one object's orbit is changed it is unlikely to collide with the other object. So, the probability of initiating a collision cascade is related to packing factor.

For intermediate packing factor values the temporal length of the cascade will depend on the number of collisions that eventually occur. In addition, the time of maximum collision occurrence will be related to the total number of collisions and how long the sequence lasts.

References

Akiba, Tatsuya, Selah McIntyre and Ann-Marie Madigan, 2024, "Tidal Disruption of Planetesimals from an Eccentric Debris Disk Following a White Dwarf Natal Kick," ApJ L, 966:L4, 2024 May 1.

Gary, B. L., S. Rappaport, T. G. Kaye, R. Alonso and F.-J. Hamsch, "WD 1145+017 Photometric Observations During Eight Months of High Activity," 2017, MNRAS, 465, 3267-3280. arXiv  

Rappaport, S., B. L. Gary, T. Kaye, A. Vanderburg, B. Croll, P. Benni & J. Foote, 2016, "Drifting Asteroid Fragments Around WD 1145+017," MNRAS, arXiv:1602.00740

Acknowledgment

The ideas described on this web page are derived from many articles, and also dinner table discussions with my friend Tom Kaye.

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