Collisions as the Source for White Dwarf Transiting Dust Clouds
Bruce L. Gary, Last Updated 2021.06.10

This web page addresses a poorly understood aspect of how collision dust clouds work.

Introduction

There are three known white dwarfs (WDs) with transiting dust clouds. For two of them the dust clouds appear to be in circular orbits: WD 1145+017 and J0328. The discovery paper for the first of these proposed that the dust clouds were produced by sublimating minerals. This was a reasonable suggestion since the temperature of the planetesimal associated with the dust clouds was extremely hot, due to its closeness to the WD. Shortcomings slowly accumulated for this dust production mechanism, and with the discovery of dust clouds orbiting the latest discovered WD there is additional concern that sublimation may not be the cause for dust clouds orbiting WDs. This is because at a distance where the J0328 asteroid source is orbiting temperatures are too cool for minerals to sublimate and too hot for volatiles associated with comets. There should therefore be new interest in the collision model as an alternative for generating dust clouds. This web page addresses a poorly understood aspect of how collision-generated dust clouds work.

Links Withing this Web Page

    Celestial mechanics  
    Dust cloud shape evolution  
    Debris disk and parent fragment  
    Specific collision case study     
    Interesting properties of dust clouds produced by a collision  
    Observational evidence  
    Conclusions  
    Further reading  
    References  

1. Celestial Mechanics - Vertical Motion

Let's begin with an imaginary "explosion" of a rock in orbit just above Earth's equator. Before the explosion the rock was traveling with velocity V oriented parallel to the equator. Consider a particle that undergoes a delta-V during the explosion in the upward direction (i.e., parallel to the rotational axis). It is in a new orbit determined by it's trajectory following the explosion. This orbit will be inclined by angle alpha = atan (delta-V / V). It's ground track will be just like satellite tracks that we've all seen:


Figure 1a. Path over earth of particle #1 if delta-V = V.

Now consider a particle with delta-V of the same magnitude but oriented in the opposite direction:


Figure 1b. Path over earth of particle #2 if delta-V = - V.

Here's the path of both particles:


Figure 1c. Path over earth of particles #1 & #2 if delta-V = V.

Notice that the two particles start out going away from each other, reaching maximum separation 1/4 orbit later, then come together at the opposite longitude of the collision, followed by another separation and coming together back at the original explosion site.

An explosion will have many particles with a range of delta-V. Here's what a few particles could do:


Figure 1d.
Path over earth of many particles from same explosion.

The cloud of particles can be described as expanding and contracting twice per orbit. For this example the maximum size of the cloud is ~ RadiusEarth. The properties of different size clouds (different delta-V max) will be the same.

Celestial Mechanics - horizontal Motions

Let's change terminology and use the term "collision" instead of explosion; this is appropriate since both produce ejecta in all directions (isotropic). Let's also shift to the site of a WD, instead of the Earth. A collision will produce delta-V in other directions than up and down. Radial components will produce eccentric orbits. They will have the same orbital period since no angular momentum was lost or gained by these inward/outward components. Here's an example of new orbits caused by inward and outward delta-V, using the WD1145 case for illustration. The view is from above a pole, looking down on the orbit plane.


Figure  2a.
Orbits for ejection in direction perpendicular to orbital motion, inward (toward the central object) and outward, assuming delta-V = 24 km/s (exaggerated for illustration purposes from a more realistic 1.5 km/s). Ejected debris expands and contracts twice per orbit.

The forward and aft components will produce eccentric orbits with different periods, since angular momentum is gained and lost by these delta-V components. This is illustrated in the next diagram (using WD1145 for illustration).


Figure  2b. Pole view of orbits for ejection in direction of orbital motion (forward and backward) assuming delta-V = +/- 8 km/s (exaggerated for illustration purposes from a more realistic 1.5 km/s). Ejected debris expands and contracts only once per orbit. Since periods differ the debris in different orbits won't return to the collision site at the same time; this will produce dust cloud smearing along the orbit.

Initially, after the explosion, the dust cloud will expand to maximum size after 1/4 orbit, then collapse to a small oblong (long in the radial direction) at the opposite longitude, followed by another expansion and contraction, ending back at the collision site as a tiny cloud of particles with their original velocities.

The previous two diagrams exaggerate the differences in orbits because they are used to illustrate a concept. Here are two counterpart diagrams that are realistic since I've adopted maximum speeds of 1.5 km/s.


Figure 2c. Locations are shown at 5-minute intervals for orbits of particles ejected in radial directions toward and away from the WD. A sense of the cloud length can be seen by the separation of corresponding points. For example, the cloud is spread out the most on the left side, opposite the collision location. The periods are the same since no orbital momentum changes are involved. 


Figure 2d. Locations of particles ejected parallel to orbital motion with delta-V = 1.5 km/s, shown at 5-minute intervals. The periods differ, and the particles arrive back at the collision location at different times. The "aft-ejected" particles begin to go ahead of the "fore-ejected" particles at ~ 1/4 orbit. At the half-orbit region the aft lead is ~ 5 minutes. By the end of an orbit the aft particle lead is ~ 8 minutes.  

The previous two figures show how different the sizes vary with orbital phase for particles ejected parallel to orbital motion and parallel to the radial direction. This causes dust cloud stretching that is episodic for one direction and accumulative for another. The next figure illustrates this.


Figure 2e. Size of dust cloud in the orbital plane vs. time for an adopted 1.5 km/s maximum ejection speeds. The blue trace shows stretching along the orbit ("orbit shear") that increases approximately monotonically. The red trace shows size variations in the radial direction, which is sinusoidal and confined to 3.6 R_wd (diameter), or 1.8 R_wd with respect to a circular orbit. The scale on the right is in units of dip length, from ingress to egress.  

Dips are never observed to be longer than ~ 30 minutes long (ingress to egress), so something must be removing particles before ~ 1.5 orbits. Usually, this removal occurs during the first 1/2 orbit. This is one of the WD1145 mysteries!

One answer to the "orbit shear" mystery that is tempting to consider is to adopt ejection speeds lower than 1.8 km/s. However, we need these high speeds to account for observed dip depths. These speeds cause vertical displacements (see Fig. 1d) that reach WD latitudes of 20 to 40 degrees. Optically thick dust clouds that are stretched out along the orbit by several WD radii have to be this large in latitudinal expanse in order to produce the observed dip depths of 30 to 60 %.

3. Dust Cloud Shape Evolution for 400 m/s Maximum Injection Velocities

The following sequence of figures show dust cloud size and shape vs. time for a full orbit. It is assumed that dust is ejected "isotropically" (within the orbit plane) with maximum speeds of 400 m/s. This is a 2-D model; particle motion perpendicular to the orbit plane is not modeled. The source for particles is orbiting counter-clockwise. All views are looking down (pole view).


Figure 3a. This orbit was obtained by placing a test particle at x,y location +95.1, 0.0 (Rwd units), specifying a velocity component in the upward direction of 315.17 km/s and zero component in the horizontal (radial) direction. The central star is WD1145 and was assigned mass = 1.19e30 kg. At equal time increments of 3 seconds acceleration was calculated and the associated velocity increment vector was added to the previous time step velocity to obtain a new velocity vector. Small adjustments of the initial velocity were made in order to achieve a circular orbit.

Vertical displacement versus orbit phase in shown n the next graph:


Figure 3b. Vertical displacement of 400 m/s particles vs. orbit phase. The phases A, B, C and D correspond to orbit locations in the previous figure.

The displacements in the orbit plane, in x.,y coordinates, was modeled by specifying an initial x,y location and assigning delta-V components of 400 m/s in the 4 cardinal directions (up, left, down and right). The following sequence shows the location of these 4 particles at times corresponding to 1/4 orbit locations (referenced in the above figure as locations A, B, C and D). Location D is a full orbit after the simulated "collision." The 4 particle locations for each 1/4 orbit interval define an ellipse within which all slower moving particles should be found. The ellipses expand and contract, change shape and rotate counter-clockwise. In the following sequence the orbit motion is always to the left, and the WD is always in the down direction.


Figure 3c. At this orbit location, A, the cloud of particles define an ellipse with maximum length ~ 0.5 Rwd and oriented at 45 degrees with respect to the orbit (and rotating in the counter-clockwise direction).


Figure 3d. At this orbit location, B, the cloud of particles define an ellipse with maximum length ~ 2.4 Rwd that is oriented at almost parallel to the orbit (and continues to rotate in the counter-clockwise direction).


Figure 3e. At orbit location C the cloud defines an ellipse with maximum length ~ 4.5 Rwd and is oriented at almost parallel to the orbit. 


Figure 3f. At orbit location D, after one full orbit, the cloud of particles define an ellipse with maximum length ~ 4.5 Rwd that is oriented exactly parallel to the orbit. (The ellipse width is actually much narrower than depicted.)

This sequence shows more clearly that Particle 1, for example, is always farther from the WD than the source. It also is lagging the other particles when the dust cloud completes an orbit by returning to the collision location. These behaviors make sense because Particle 1 was ejected in the forward direction, so it went into a larger orbit, with a longer period.

It should be noted that during the orbit depicted above, there were particles not included in this model that were ejected in the up and down directions (perpendicular to the orbit plane), referred to as motion in the z coordinate. Their distances from the orbit plane underwent two cycles of expansion and contraction. For example, assuming 400 m/s ejection speeds away from the orbit plane, at the time depicted by "A" the z location was ~ 1030 km, or 0.12 Rwd. This corresponds to a blockage 15 % of WD light (assuming delta-V occurs in both directions, and dust cloud is stretched out to at least a WD diameter). Clearly, greater values for delta-V are required.

The diagrams above (with maximum ejection speeds of 400 m/s) is apparently typical for WD1145, because 15 % dip depths are common. However, greater ejection speeds are needed to account for the deeper dips. We need to consider maximum ejection speeds of > 1 km/s in order to account for observed dip depths as large as 65 %.

Maximum Ejection Velocity of 1.8 km/s

The next graph shows the relation between ejection speed and dip depth. It assumes that the dust cloud is stretched along the orbit many times Rwd, forming a horizontal band that passes through disk center. 


Figure 3g. WD disk blockage vs. ejection speed, assuming an optically thick dust cloud that is greatly stretched by "orbit shear" (i.e., broad band along orbit).
 
Since dips have been observed with depths of 65 % we have to assume that ejection speeds can be as large as 1.8 km/s. Sublimation models are thought to be limited to a feww 100 m/s maximum speeds. That's the main reason collisions are attractive.

4 - After Many Orbits

As shown in Fig. 2e the "along orbit size" of the dust cloud continues to lengthen after the first orbit. After 6 orbits the dust cloud is 100 R_wd in length. The circumference of the orbit is 600 R_wd, so the two ends of the dust cloud will come together after ~ 36 orbits, or one week! 

A more detailed calculation of orbits for particles ejected isotropically has been performed by Jackson et al. (2014). Here's a graph from that paper showing a parameter related to particle density (projected onto the equatorial plane) after so many orbits that a steady-state of particle density is achieved.


Figure 4. A figure from Jackson et al. (2014) showing a parameter related to density of particles after many orbits of an isotropic ejection of particles from a single collision.The original collision site is on the right.
 
According to the above graph by Jackson et al. (2014) the greatest probability of additional collisions after the initial one is located at the original collision site. Note that after a steady-state is established (when "orbit shear" no longer changes the shape of where debris is found) the debris disk has a volume that is fixed in inertial space. The thickness of the debris disk is maximum at the 1/4 and 3/4 orbit azimuths e.g., one R_wd), and thickness is very small at the collision and opposite azimuth. The minimum of thickness at the 1/2 orbit azimuth produces a high density of debris and accounts the line of high probability of collision in the above figure.

All the preceding description is for a single collision that produces an isotropic ejection of particles that spread out to form an asymmetric debris disk. We are now ready to consider one crucial new aspect of the model: what if the original fragment survives the original collision and continues to orbit within the debris disk?

Fate of a Large Fragment that Produces a Debris Disk

Imagine that a large fragment is impacted by a smaller fragment and this produces isotropic ejection of particles that form a debris disk (after a week or two, assuming, for now, that we're dealing with WD1145). The debris disk will have a "particle size distribution" (PSD) that ranges from photometrically relevant (< 1 micron radius) to photometrically irrelevant (>> 1 micron) yet collisionally relevant. There is a finite probability for additional collisions during each passage of the parent fragment through the high density region of the debris disk original collision site. This probability is specified by the density of particles, the PSD function and the size of the parent fragment. If this probability is close to one, then during every orbit the parent fragment will create a new component of debris that starts out small and undergoes the expansion pattern described in the previous sections. In this way the original debris disk is continually added to, or at least replenished. Since the new component of debris begins confined to a small volume it may have a sufficient density of photometrically relevant particles to be optically thick. The debris disk, on the other hand, is so spread out that a line-of-sight through it at a random azimuth may be optically thin. Note that particles >> 1 micron are relevant for producing collisions, but are unimportant for blocking light (simply because they are much less numerous).

Keep in mind that the debris disk is a volume with a specific shape, fixed in space, with debris whizzing around at km/s speeds, and the parent fragment is orbiting within it and encountering debris with a density that varies with the same pattern during each orbit. There will in fact be several fragments orbiting within this debris disk. Over time, as the cascade of secondary collisions accumulate, the parent fragment will lose mass and shrink. With a reduced size it will be less likely to encounter debris, so the dust cloud that is kept in place around it will dissipate and it will no longer produce dips.

Consider the scenario in which the original collision yields several large fragments in similar orbits. After a debris disk is established from that collision (after a week or so) each large fragment will experience collisions within the debris disk where debris density is the greatest (site of the original collision), and each of these secondary collisions will contribute to replenishing the debris disk. If several fragments are in similar orbits they can all experience greater activity becasue they all pass through the debris disk's high density region. In this way one fragment's increase in activity can cause neighboring fragments to also increase their activity. This was observed in 2017 for WD1145 (see Fig. 7h, below). 
 
This is a mechanism that might be capable of maintaining an optically thick dust cloud with a stable dip depth and width until the fragment loses mass and shrinks in size so that it no longer has a high encounter probability. What's needed for assessing the merits of this model are detailed calculations showing quantitatively that encounter probabilities are high enough to sustain the steady-state replenishment of a parent fragment's fresh and small-sized debris clouds with sufficient optical depth to produce dips similar to those that are observed.

5 - Specific Case Study

Let's try to "quantify" a collision scenario that could account for some of the observed dip behavior. Some of my assumptions will surely need revision since I'm not expert on collisions. My purpose for this exercise is to show that a collision model can work if a large volume of N-dimensional parameter space is allowed for exploration. It will up to others, more knowledgeable on collisions and sublimation, to determine whether or not a feasible volume of this parameter space can account for observed dip behavior. My excuse for entering this unfamiliar territory is that no one else is doing it, and someone should start doing it!

I'll start with the following assumptions:
    1) the parent fragment is 10 km in radius orbiting in a 4.5-hour circular orbit around WD1145,
    2) an initial collision occurs when a 1 km object (0.1 % as massive) impacts the 10 km object which produces debris from the completely shattered smaller object
              (debris from the parent object will also be included in the debris cloud but it will not be considered in this case study),
    3) this initial collision produces debris with a size distribution of large chunks (i.e., < 500 m radius),
    4) subsequent collisions (a "cascade") populates smaller sizes, down to 0.3 micron eventually,
    5) debris has a velocity distribution that is isotropic and has a maximum speed of 500 m/s,
    6) the smallest particles have the highest speeds,
    7) the fragment and debris cloud is observed from Earth to cross the WD disk at an orbit location 3.7 hours past the collision site,
    8) sublimation occurs for particles smaller than 1 micron radius

For these assumptions Earth observers see the dust cloud when it is at an orbit position 252 degrees of azimuth past the collision site (between B and C in Fig. 3a). The vertical extent of the cloud is 78 % of its maximum extent, and provided the cloud is broader in azimuth than the WD, an opaque cloud would produce a dip depth of 14 %. This is the median depth observed during the first 3.5 years of ground-based observations (which explains the specifics of some of the above assumptions). I am not assuming that the cloud is opaque on the first orbit; more things have to happen to produce a high opacity. Here's the extent of the cloud 3.7 hours after the initial collision.


Figure 5a. Extent of dust cloud 3.1 hours after the initial collision.

Keep in mind that the above figure shows the extent of the smallest particles. Large particles, which I will refer to as debris, will be confined to smaller volumes. Since its the small particles that dominate optical depth (e.g., < 5 micron radius), any dip that is produced by this cloud will have a length (ingress to egress) corresponding to the outer limits of the dust cloud, which for the above case would be 5 x R_wd / 300 km/s = 9 minutes (assuming high opacity throughout the cloud). So far, all these cloud properties are compatible with dip observations.

Let's estimate the optical depth of the the dust cloud at the 3.1-hour time of its evolution. To do this we have to adopt a "particle size distribution" (PSD) function. Instead of using an analytic model, since this is just a demonstration of approximate feasibility, I'll treat particles as belonging to one of 10 size bins, as illustrated in the next figure.


Figure 5b. Debris radius histogram after 3 and 10 collision "steps."

In this PSD model it is assumed that we begin with one object with a radius of 1000 m at time step 0. A time step interval is defined to be how long it takes for half the number of objects in a size bin to be converted to the next smaller size bin. Therefore, during the collision cascade each size bin loses objects to the next smaller bin and gains a smaller number of objects from the next larger bin (if objects are present there). The log/log slope of the steady-state histogram is close to that observed for Saturn's A-ring: N(r) = c r-11/4.  As cascades continue the right end of the function decreases, meaning that the original 1 km object eventually is worn down to nothing - which eventually happens to the large chunks created by the initial impact.

This oversimplified model has the unrealistic assumption that one collision produces only fragments in the next smaller size bin. In reality, a fraction of a second after an initial impact faster moving fragments will be colliding with slower ones, so in fact one impact unleashes a cascade of smaller collisions as the debris leaves the initial impact site. Therefore, all size bins can be populated after the initial impact. The procedure used to create the above histogram is just a mathematically convenient way to determine the shape of a histogram after some form of steady-state is achieved, which might happen after just the initial impact.


After the cascade of collisions has led to a steady-state (all size bins populated and the largest size bin is still populated) it is possible to calculate the projected area of all particles. This is shown in the next graph: This figure illustrates why the smallest particles dominate blockage of WD starlight.


Figure 5c. Projected area of all particles in specific size bins, assuming no overlap. 

The most important size bin is 3 microns, which has a projected area for all particles in that bin = 2e13 m^2. For comparison, the projected area of a WD1145 hemisphere = 11e13 m^2. The sum of projected area for the 3 micron, 30 micron and 300 micron bins (particles within the size region 1 micron to 1 mm) equals 3.8e13 m^2. This is sufficient for producing a 35 % depth dip (if there was no overlap of particle areas). It is conceivable that when considering particle overlap a 14 % dip could be produced by the scenario under consideration. Notice that all debris larger than ~ 1 mm is irrelevant for producing dips. Their projected area (for this model) is a mere 9.4e11 m^2 (or < 1 % of WD1145 hemisphere). You'll notice that I omitted inclusion of the 300 nm particles in my calculation of projected area for the dust cloud. This is because these small particles have been shown to rapidly overheat and sublimate out of existence (Xu et al., 2017).

At the time corresponding to Fig. 5a, after less than one orbit following the original collision, according to this model the PSD will consist of only large debris; i.e., no debris as small as 0.3 micron. Therefore, this model predicts no dip depth for the first orbit through the Earth's line-of-sight (LOS) to the WD. A more realistic model would produce some debris at all sizes, but probably very small amounts at 0.3 micron, meaning that during the first orbit following an initial collision there should not be an observable dip.

Assumption 6, above, states that only the smallest particles have velocities as great as 500 m/s. I have no idea how to model the function for collision ejection speed vs. particle size, so I've decided to consider the possibilities shown in the following figure:


Figure 5d. Arbitrarily selected functions describing ejection speed vs. particle size, using the equation V = constant  / (radius ^ exponent). 

Note that if all particles had the same kinetic energy the speed of 30 m chunks would be an astoundingly low 1e-7 m/s (assuming 500 m/s for 1 micron particles), so I reject this assumption for guidance in adopting a model for velocity vs. particle size. Since particles are colliding with each other after the initial impact there will be an effective "viscosity" which will cause faster particles to lose speed and slower particles to gain speed. I have no idea how to model this situation, and maybe the best way to learn about this is from laboratory experiments. Since I'm not aware of the results of such experiments I will adopt an arbitrary model. A first step is to adopt one speed for each particle size. This overlooks the more realistic case in which each particle size exhibits a histogram for speed (e.g., Maxwell-Boltzmann distribution, or something similar). The above chart is intuitively reasonable and I have adopted the middle model (exponent = 0.2). Accordingly, during the debris cloud's first passage through the Earth's LOS most of the debris won't be as spread out as shown in Fig. 5a. For example, the 30 m chunks will be moving with a speed of ~ 30 m/s whereas the 1 micron particles will be moving with speeds of 500 m/s. This means that the large chunks will have a rate of expanding along the orbit ("azimuth shear") that is only ~ 6 % of that for the 1 micron particles. This is important because it means we can account for the presence of dips that don't change their width or depth for weeks by viewing the large chunks as continuously producing small particles which have a finite lifetime, as will be discussed below.

Referring back to the histogram of debris size (Fig. 5c), after a steady-state is established half the mass of the debris cloud will be in chunks with a size greater than 3 cm, traveling with speeds less than 60 m/s. This population of large debris can be viewed as a source for future dust particle production. The following figure shows how confined in space this large debris population is.


Figure 5e. Extent of small particle dust cloud (< 3 micron) and most massive half of debris (> 3 cm), 3.1 hours after the initial impact.

It's possible that during the dust cloud's first passage through the Earth's LOS, 3.1 hours after the collision, there could be a sufficient number of small particles to produce an observable dip. If so, then the shape of the dip would be determined by the velocity distribution versus particle size. Let's predict what our simple model so far predicts for optical depth versus distance from the large (10 km) fragment. So far my model assigns just one speed for each size bin. This exercise will therefore provide a way (in theory) to derive a speed histogram vs. size. One candidate for a speed function is the Maxwell-Boltzmann distribution, shown below:


Figure 5f. A Maxwell-Boltzmann distribution histogram for speeds with a "tail" near 500 m/s (corresponding to 3 micron particles in the above model). 

For larger particles their speeds will be slower, as shown in the next graph:


Figure 5g. Maxwell-Boltzmann distribution histogram of speeds for particle sizes 300, 30 and 3 micron (assuming the relation in Fig. 5d).

Consider the fact that the number of 3 micron particles is ~ 250 times the number of 30 micron particles (in Fig. 5b). Another way to illustrate this big difference is shown in the next figure.


Figure 5h. Speed histogram for 3 particle size bins.

Since a particle in one size bin obstructs 100 times the area of a particle in the next lower size bin the importance of particles versus size is better portrayed in the next graph.


Figure 5i. Total area blocked by particles per increment of speed, for three size bins.

As already noted, in terms of blocking WD starlight the 3 micron particles (size range of 1.0 to 10 microns) are the most important, with less importance for each larger size bin. But what's the number density of particles versus distance from the 1 km asteroid that's near the center of the dust cloud after a specific time, such as 3.1 hours after the collision? If we begin with an assumption of spherical symmetry (no Keplerian azimuth shear) this is easily calculated by assuming that particles are distributed in a series of concentric shells whose radii are set by the speed of the inner and outer boundaries for the size bin under consideration. For example, at time t seconds after the collision the shell containing 3 micron particles traveling at 100 m/s extends from radii = t 90 m/s to t 110 m/s. The density of particles in that shell is simply the total number of particles in that size bin and within that speed range (Fig. 5h) divided by the volume of the shell (4 pi r2 delta-r). The light blocking power of this subset of particles is obtained by multiplying 1) the number density by 2) the cross-section of each particle by 3) the projected area of the volume cell under consideration. Figure 5i is a convenient combination of parameters that can be used for this task.

It will be helpful to calculate, for each size bin, particle number density vs. radial distance from the center (e.g., from the parent asteroid, with radius 10 km). This will entail calculating shell radii boundaries (using Fig. 5h), which determines the shell's volume. Knowing the total number of particles within this volume enables calculation of particle number density within that shell. This is the particle number density that we seek.


Figure 5j. Number density vs. radial distance from the large asteroid at the center of the debris cloud, for 3 size bins.

What we really need is the projected area of the particles per cubic meter vs. radial distance from center. This is shown in the next graph.


Figure 5k. Projected area of particles in a cubic meter vs. radial distance from center, for 3 size bins.

Finally, here's the sum of projected area, per cubic meter, for the 3 size bins we've considered vs. radial distance from the cloud center.


Figure 5l. Projected area per cubic meter of the three size bins added together vs. radial distance form the center.

With this knowledge we can now consider a way to create a light curve dip for a hypothetical observer who sees the above-described cloud pass through his line-of-sight. We shall perform this calculation using a cube of cells that includes the spherical dust cloud at t =3.1 hours. For each cell, with x,y,z coordinates, we calculate distance from the center, which allows for a determination of the projected area of the three particle sizes (by simply multiplying the value from the above figure times the volume of the cell).


Figure 5m. Shape of light curve dip for the above modeled dust cloud, using size bins: 3, 30 & 300 microns.

Question: We know that essentially all observed WD1145 dips can be fit by an "asymmetric hypersecant" (AHS) function, so can the model dip be fit by an AHS function?


Figure 5n. Asymmetric hypersecant (AHS) function fit to the model dust cloud dip.

The answer is "Yes, the model dust cloud's predicted dip shape can be fit by the AHS function!"

The collision model predicts a very short dip length when the cloud is observed only 3.1 hours after the original collision. Whereas total dip length is ~ 4 minutes the FWHM length is 1.2 minutes. If this dust cloud had been observed somewhat later, at the 3.65-hour time after the original collision (same vertical extent), the FWHM dip length would have been 1.5 minutes. Such short dips are rare but they have been observed when large aperture telescopes with high SNR have been used (e.g., Croll et al., 2017, Gaensicke et al., 2016). I conclude that original collision events are capable of producing a mature, or steady-state distribution of particle sizes. In other words, it is possible to observe a dust cloud during its first orbit following a collision. However, most observed dips are broader and require more than one orbit to produce the typical 5 or 10 minute FWHM lengths. This topic will be addressed below.

So far we've seen that a collision with the adopted assumptions can produce a dip that resembles some observed dips after less than an orbit following the original collision. The model assumption that makes this possible is that during the original collision there are a sufficient number of secondary collisions (during the seconds following the original collision) to populates all bin sizes in a way that resembles a steady-state.

The next goal is to assess the feasibility of the 10 km fragment undergoing secondary collisions every time it passes through the original collision site; I will refer to this location as "the funnel."

Here's the view of the dust cloud after exactly one orbit, when the parent fragment and all debris has returned to the original collision site.

 
Figure 5p. Two views of extent of small particle dust cloud (< 3 micron) and most massive half of debris (> 3 cm), 4.5 hours after the initial impact (when the 10 km parent asteroid fragment and all debris returns to the original collision site after just one orbit). In the left panel the width of the dust cloud is greatly exaggerated. The small arrow pointed up has a length corresponding to the extent of the most massive half of debris. In the right panel, which is a90-times magnification of the left panel, the length along the orbit of the most massive half of debris extends well beyond the graph borders (+/- 0.2 R_wd). In both panels the parent 10 km asteroid fragment is represented by a circular symbol.

The radial width of the dust cloud edge is very small, e.g., 0.00034 R_wd (2.8 km) for the crude calculation of this analysis (time-steps of 5 seconds). Theoretically it should be zero, and it would have been if the accuracy of my celestial mechanics had been perfect. However, this zero width assumes that there were no secondary collisions during the seconds following the original collision. I haven't been able to include these secondary collisions in this model so the width at the funnel will have to be assigned arbitrary values. For example, if the last of these secondary collisions occurred 1 second after the original collision then the loci of orbit origins for all particles will be confined to a sphere 500 m in radius, and this would determine the width of the dust cloud at the funnel location.

Let's think of the funnel as a section of a ring of length 4 R_wd with a circular cross-section having a radius that is somewhere within the range 100 m to 4 km. Prior to the dust cloud's arrival at the funnel particles will be "collapsing" toward the funnel ring (i.e., orbit) with speeds the same as they left the original collision site one orbit earlier. At some specific time before reaching the original collision site we can imagine a snapshot of the locations and velocities of the small and larger particles, which is shown in ht next figure.

  
Figure 5q. Geometry of particles collapsing to original collision site (where the 10 km fragment is located) showing how the smallest particles are coming in from greater distances and at greater speeds than the larger particles. Left panel shows what a symmetric pattern of velocity vectors would look like, while right panel shows velocity vectors that result from secondary collisions concurring during the few seconds following the original collision.

Keep in mind that these particles are collapsing like shrinking tubes aligned with the orbit, and the parent fragment is located at just one position along the tube's center line. In calculating the probability of a collision of a particular particle with the 10 km parent fragment this will be important.

In addition to collisions of particles with the parent fragment there can be collisions of particles with each other. Both collision types have to be considered when evaluating the ability of funnel passages to replenish dust lost to sublimation during the previous orbit. The easiest collision type to calculate is debris/parent fragment, so that's waht I'll treat next.

Notice that the width of the dust cloud stated above, 500 m or less, is much smaller than the size of the parent asteroid fragment of 10 km. This means we can assume that all particles will collide with the 10 km parent fragment when the particles cross the orbit plane when the 10 km parent fragment is at the same orbital azimuth. By same orbital azimuth is meant same to within 10 km. For example, consider the 3 micron particles. They cross the orbit plane when they are stretched out along the orbit a distance of 5.5 R_wd (diameter, using the 500 m/s particles for defining cloud size). A particle density-weighted distance would be ~ 3 R_wd (total length), or 1.5 R_wd (radius), which is 1.5 x 8.4e+6 m = 1.2e7 m. The 10 km (radius) parent asteroid fragment represents a fraction of 1e-3 of this cloud's length. In other words, only ~ 0.1 % of the 3 micron particles collide with the 10 km parent fragment.

Using this approach for the other particle size bins yields the following figure:


Figure 5r. Fraction of particles colliding with the 10 km parent fragment after one orbit as a function of particle size.

About 3 % of the 300 m debris will collide with the 10 km parent asteroid fragment after one orbit. These collisions will add to the number of particles in the 30 m size bin. This is a small change, and since the percentage additions for all other sizes are even less impressive we can see that "particle/parent" collisions won't be an important contributor to dust cloud longevity. 

Next let's consider "particle/particle" collisions.















Temporary halt in this modeling project!
 



6 - Interesting Properties of Collision-Created Dust Clouds

[This section needs revisions related to the insights described in the previous section.]

A collision resembles an explosion. Ejected particle directions will be approximately isotropic for both. The range of speeds will have a maximum, and everything smaller should exist. A collision in orbit will therefore be followed by the same behavior described above.

Consider what might happen at the times of minimum size. Particles (and larger fragments) are likely to collide, and could produce a new population of smaller particles. A quasi-continuous production of new particles may occur twice per orbit. A particle size distribution will eventually be achieved. The total absorbing and scattering power of the dust cloud at a given phase could therefore vary: a steady decrease due to orbit shear plus occasional injection of a new population of particles during the contraction phases.

If particle injections are isotropic then some will be in shorter period orbits and some will be in longer period orbits. These will be the particles that had an ejection velocity component in the forward or afterward directions. Because of there being a range of periods for the particles the shape of the dust cloud will spread out along the orbit more each orbit. The shape will evolve from spherical shortly after the collision to an ellipsoid with a circular cross-section. The length of the ellipsoid will increase with time. I'll refer to this process as "orbit shear."

Observations of dips show the effect of orbit shear during the course of a few days after a dust cloud forms. However, the stretching out along the orbit reaches a maximum amount and the length of the dip remains approximately constant for many more orbits (weeks, o until the dip disappears). What can limit the stretching caused by orbit shear? We don't know, but we can speculate. I can think of two candidates for limiting orbit shear stretching: 1) particles in smaller orbits (with periastrons closer to the WD) will be hotter, and may sublimate. This could limit the length of the ellipse in the forward direction (ingress). 2) particles in larger orbits (apoastrons father from the WD) will be in the way of dust clouds generated by the planetesimal in the next farther out orbit. For example, WD1145's inner-most planetesimal, in the "A" orbit, is at a distance from the WD of 97.6 R_wd. The nearest neighbor planetesimal is in the "D" orbit, at 98.4 R_wd. In the rotational reference frame of the "A" system the "D" dust clouds are moving past the "A" dust clouds in an outer orbit (in the backwards direction). Those "D" dust clouds will be expanding and contracting, just like the "A" dust clouds. Any "A" particles that venture out far enough may encounter an expanded "D" dust cloud; when this happens it will have its orbit altered due to impacts with "D" dust. Ironically, the "D" dust will impart a forward delta-V to the "A" particle, and cause it to be in an even larger and longer period orbit. In the extreme case the "A" particle could be "entrained" by the "D" dust cloud. This would remove it from the ellipsoidal "A" cloud, and lead to a limit on how far in the aft direction (egress) the ellipsoid could extend.

I haven't worked out numbers for the effects described in the previous paragraph, so this is so far just speculation on why orbit shear has a limit on how stretched out (along the orbit) a dust cloud can become. Those effects are destructive, in the sense that the "A" dust cloud is losing particles wthat had been ejected with too great a component of forward or backward velocity component. This particle loss of a dust cloud is compensated by particle production that occurs at each compression cycle. The two processes, particle loss rate and particle production rate, should lead to a steady-state size for the ellipsoidal dust cloud. We need such a result in order to be in agreement with observations (next section).

Note that if the cloud is detected by its absorption and scattering of light from a central source (a WD) the observer's longitude will determine what phase the cloud is at each time it transits the central object. The observer will see the cloud at the same phase, every time, and won't know whether the cloud is much larger or smaller for other hypothetical observers at other longitudes.

A collision could produce gas as well as particles. The gas will expand and contract synchronously with the cloud. The gas will have a distribution of velocities along the line-of-sight, which wil vary with phase. At the maximum expansion phase the distribution of radial velocity will be close to zero width. The distribution of velocities will be maximum when size is minimum. If WD light is absorbed by the gas it is theoretically possible to determine which phase is being observed. There will be an ambiguity, of course, except at maximum and minimum phase when the radial velocity spread is minimum and maximum, respectively.

Notice that the cloud's optical depth will vary in opposite manner with size. The same cloud could be physically small and optically thick for an observer at one longitude, and the opposite for an observer at another longitude. Earth-based observers may think that a shallow dip is for a small cloud, but we have no way of knowing that the same cloud might be viewed by some alien civilization as large with a deep dip depth.

If both gas and dust are present in a cloud the "viscosity" of the cloud will vary with a cycle of half an orbit. Maximum viscosity will occur when the cloud is smallest. This might inhibit the effect of orbital shear by altering the velocities of the particles with extreme relative velocities.

Dips that appear suddenly, with no evidence of any dip activity prior to the start date, can be easily explained by this collision model. A "turn-on" can occur whenever there's an impact of two significantly massive chunks.

A collision might not create a dust cloud initially, but instead create just two or more chunks that themselves can later become the source for dust clouds. The daughter chunks will be in different orbits, and will drift with different slopes in a waterfall diagram. Each daughter chunk can have a different "turn-on" time, depending on when other pieces in close orbit with a neighbor chunk impacts the main chunk (during a minimum size phase), or when other smaller pieces that accompany the main chunk impact each other. For example, if two chunks are created from a larger one due to a collision, there is a finite probability that they will impact each other during compression phases. Eventually, when the chunks impact, they will create a dust cloud that is observed as a dip turn-on.

If collisions have a characteristic impact speed there will be a characteristic profile of velocities for the fragments and particles. The most interesting parameter for this scenario is the maximum velocity of particles. This will determine how large the dust cloud will be at each of its maximum expansion phases. If, for example, the WD1145 delta-V maximum is 1 km/s, the maximum size of the cloud will be 2700 km (radius). During the maximum size phase such a dust cloud will be a sphere of radius 2700 km, and after orbit shear has spread out the cloud in the orbit direction it will be an oblong with a cross-section of 2700 km radius and many times this in length. WD1145 has a diameter of 8550 km, so if the dust cloud crossed the center of the disk, and if the dust cloud was optically thick throughout, the cloud could cover as much as 35 % of the disk area and reduce starlight by the same amount. Since we often see dips this deep the 1 km/s model is appropriate to consider. This thought will be developed in the next section.

7 - Observational Evidence

Is there evidence for collisions? Yes, and mostly it comes from waterfall plots. Consider the following examples.

 
Figure 7a. An 8-month waterfall plot for WD1145 showing one collision event with possibly 4 to 6 dust clouds following different drift lines. Notice that "turn on" can take up to 80 days (420 orbits).


Figure 7b. A collision on Day 416 created 4 fragments that "turned-on" 3 or 4 days later.


Figure 7c. Waterfall diagram suggesting that WD1145 underwent a collision event on 2021 Feb 17.

My last example of a collision found in a waterfall drift line divergence pattern comes from the very first waterfall plot created from ground-based observations (Rappaport et al. 2016).


Figure 7d. These drift lines diverge from a date in August, 2015, which corresponds to the time dip activity level for WD1145 increased dramatically, by a 100-fold, compared with ground-based measurements made 5 months earlier and before that by the Kepler spacecraft.

An indirect way to check for consistency with collisions is to compare histograms of dip depth with collision model predictions.


Figure 7e.
Histogram using uniform intervals for log(depth).

In the above histogram depths < ~ 7 % are under-counted due to SNR limitations of my telescope (14"). The break at 40 % is real and may be related to the inclination of dust cloud orbits that produce a preference for transiting in front of one hemisphere of the WD disk more than the other hemisphere.It could also be related to the maximum dust particle velocities produced by collisions 9e.g., ~ 1.5 km/s). A crude comparison of the observed flat slope is in agreement with a model developed by Kenyon and Bromley. More work is needed on this matter.

According to Kepler data WD1145 has dips with 6 different periods, ranging from 4.5 to 4.9 hours. Consider the most active period of 4.5 hours, which corresponds to a circular orbit radius of 97.6 R_wd. If dust clouds with a cross-section radius of 0.35 R_wd are common (as derived in the previous section), and if they are usually at the 97.6 R_wd orbit location, they could "scour," and keep clean, less frequent dust clouds throughout the region 97.6 +/- 0.35 R_wd, or from 97.2 to 98.1 R_wd. In other words, we should expect to find that dip periodicities have a characteristic spacing that is defined by the typical maximum collision velocity for planetesimals. The next figure is evidence that supports this speculation.


Figure 7f. The vertical lines correspond to the 6 Kepler periodicities. Notice the approximately uniform spacing. (The "missing" period corresponding to a distance of 100.5 R_wd may in fact be present, but at a level undetected at the time of the Kepler observations due to temporary inactivity.) 

Based on the spacing it could be speculated that maximum collision speeds are ~ 1.5 km/s. If the dust clouds produced by these speeds are optically thick when they are at their maximum size, then dip depths of ~ 60 % should occur (and they do).



Figure 7g.  Dust cloud width and depth vs. date for a 3-month interval in 2019, for one specific dip.

This example may be what was described in the previous section where orbit shear was discussed. Here's my scenario for events: 1) An impact of two chunks in the "A" orbit occurred on about Day 72. A spherical dust cloud rapidly expanded (during a 1/4 orbit) to a size (and optical opacity spatial function) that obscured a few % (<10 %) of WD light. The size of the cloud must have been large, ~ 0.05 phase units (tau1 + tau2). This size corresponds to ~ 15 times the diameter of WD1145 (stretched out along the orbit by "orbital shear"). 2) During each contraction (every 1/2 orbit) the cascade of collisions continued and produced more dust in the cloud. The optical depth of the cloud was increasing even while it was decreasing in size. Why was it decreasing in size? Because the particles with the farthest out apoastrons were entrained by the "D" dust clouds, and were removed from the following edge of the "A" ellipsoid" cloud. The leading edge may have been contracting due to these particles orbiting closer to the WD and becoming hot enough to sublimate. The cloud kept increasing in optical depth until Day 100, when 30 % of WD light was blocked. 3) From Day 100 to the end of observations there was a steady-state production of dust at minimum size times (when speeds and number density of chunks were maximum) balanced by loss of particles to entrainment by the "D" dust clouds and sublimation of the particles that orbited too close to the WD.


Figure 7h. An unusual case of a wide range of phases undergoing simultaneous increases in activity, followed by a coordinated decrease in activity.

Conclusion

Some evidence exists for collisions as a mechanism that produces transiting dust clouds for WD1145.

Further Reading

Here's Saul Rappaport's derivation of equations to be used for calculating key parameters for orbits of particles emitted from a source that is in a circular orbit.



The planetary system formation community embraces collisions, as exemplified in the following article:



This paper describes the celestial mechanics of the collection of collision particles expanding and contracting twice per orbit (as worked out by co-author Alan Jackson).

The arXiv preprint of this article is available at link 

References   

Jackson, Alan P., Mark C. Wyatt, Amy BOnsor and Dimitri Veras, 2014, "Debris from Giant Impacts Between Planetary Embryos at Large Orbital Radii," MNRAS, 440, 3757
Xu, S., S. Rappaport, R. van Lieshout & 35 others, 2017, "A dearth of small particles in the transiting material around the white dwarf WD 1145+017," MNRAS link, preprint arXiv: 1711.06960

Croll, Bryce, Paul A. Dalba, Andrew Vandenburg and 12 others, 2017, "Multiwavelength Transit Observations of the Candidate Disintegrating Planetesimals Orbiting WD 1145+017," ApJ, 836:82 (pdf)

Gaensicke, B. T, A. Aungwerojwit, T. R. Marsh + 11 others, 2016, "High-Speed Photometry of the Disintegrating Planetesimal at WD 1145+017: Evidence for Rapid Dynamical Evolution," arXiv 2015 link  

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