Moon 13 degrees away, so noisy! Also, all high air mass.
2019.07.05 - BG16
2019.07.04 - BG16
Windy, so noisy.
Dip labeled "3" is the DIp#1 in the waterfall plot
for the A system.
2019.06.21 - BG16
2019.06.20 - BG16
2019.06.19 - BG16
2019.06.18 - BG16
2019.06.16 - BG16
Poor data quality (wind), but available if desperate.
2019.06.15 - BG16
2019.06.13 - BG16
2019.06.09 - BG16
2019.06.05 - BG16
2019.05.30 - BG16
2019.05.29 - BG16
2019.05.28 - BG16
2019.05.27 - BG16
2019.05.26 - BG16
2019.05.24 - BG16
2019.05.21 - BG16
2019.05.18 - BG16
Nearby full moon increased noise level.
2019.05.12 - BG16
Clouds before and after a
short clearing provided ~ 1/2 orbit coverage.
But it can be combined with data from the night before to
provide ~ a full orbit of coverage.
2019.05.11 - BG16
Clouds before and after a short clearing provided < 1 orbit
2019.05.08 - BG16
2019.05.05 - BG16
2019.05.04 - TK16
2019.05.02 - BG16
2019.04.28 - BG16
2019.04.25 - BG16
2019.04.24 - BG16
Notice several events of cloud losses, as much as 1.5
2019.04.23 - BG16
This is the second observation with my new telescope. It's short
because I'm calibrating other properties of the hardware, but
the data indicates that quality is improved over what my Meade
2019.04.21 - TK16
Warm WX, and darks not at same temp as lights so cal did poor
2019.04.21 - BG16
This is the first LC I made with a new telescope,
an Astro-Tech 16" Ritchey-Chretien.
This is the first LC made with a new telescope, an Astro-Tech 16"
Ritchey-Chretien. It was windy most of the night, so data quality
2019.04.13 - BG14
2019.04.11 - BG14
2019.04.09 - BG14
2019.04.08 - BG14
2019.04.03 - TK32
2019.04.03 - TK16
2019.04.03 - BG14
Wind degraded several images. A few clouds were also
2019.04.02 - TK16
9 % outliers removed.
This LC has 9 % outliers removed.
2019.03.29 - TK16
2019.03.29 - BG14
2019.03.24 - BG14
2019.03.23 - BG
Full moon was ~ 34 deg away, causing high noise level.
219.03.15 - TK32
This is an average of data from 3 telescopes.
219.03.15 - TG16
After averaging in groups of 3.
2019.03.15 - BG
2019.03.14 - BG
2019.03.09 - TK
2019.03.09. - BG
2019.03.04 - TK
Averaging groups of 5 (after sorting all observations of this
date by phase).
Averaging groups of 3 (after
sorting all observations of this date by phase).
No averagin (a symbol for each image). Both data sets show a
brightening at phase 0.11. Could this be forward scattering from
a dust cloud that doesn't block the WD disk?
2019.03.04 - BG
2019.03.01 - TK
2019.03.01 - BG
2019.02.28 - BG
Dip #1 has changed shape since the day before: ingress is brief
and egress is slow. One of the group of 3 D dips has disappeared
(the middle one). Of these 6 dips 3 are produced by A fragment
and 3 are produced by D fragments.
2019.02.26 - TK
2019.02.26 - BG
2019.02.25 - TK
2019.02.24 - TK
2019.02.24 - BG
2019.02.23 - BG
I'm not saying the dip at phase 0.56 is the D dip, first seen
on Feb 12, because it may just be a coincidence that it is
located where we were looking for it.
A gibbous moon (81 % illuminated) was 30 degrees away, which
decreased SNR significantly. The first and last dips are 4.55
2019.02.18 - TK
A cleaned-up version.
2019.02.18 - BG
2019.02.17 - TK
2019.02.17 - BG
2019.02.16 - BG
I was "desperate" to observe that D dip, but cloudiness ruined
2019.02.12 - TK
Using the Kepler D period = 4.5500 hrs (and an arbitrary
BJD_ref) this date's phase-folded LC for both HAO and TK16 data,
shows the D dip stable in phase, at 0.79.
2019.02.12 - BG
On this date a "sharp" dip was first seen that belongs to the
D-fragment orbit. During one LC the dip appeare twice, with a
separation o 4.552 +/- 0.004 hours.
This analysis shows that with an exposure time of 60 seconds,
and a cadence of 71 seconds, the actual dip depth (assuming it's
the same for the first and second appearance) is slightly deeper
than the solution based on measurements. A source function with
a deeper depth was convolved with a rectangular exposure
function at appropriate exposure start times, and these results
are shown by the red dash symbols. The source function has a
depth of 25 % (vs. the 23 % that fits the measurements). There
is negligible affect on the apparent width of the dip caused by
a finite exposure time.
This is the AHS model fit when the measured values are fitted
(using sum chi-squares), using one depth and shape with a UT
offset for the second appearance of the dip following the first
appearance by 4.552 hours (also solved-for).
Using the Kepler D period =
4.5500 hrs (and an arbitrary BJD_ref) this date's
phase-folded LC for HAO data shows the D dip stable in
phase, at 0.79. (I later refined the D dips period to be
Phase folding with the A-fragment period shows the main dips at
The more accurate separation of the two main dips is 4.552 +/-
This is the LC that forced me to recognize that a D dip was
present, and it was "sharp" (like most dips at "turn-on").
2019.02.07 - TK
2019.02.07 - BG
2019.02.02 - TK
2019.01.28 - TK
2019.01.27 - TK
2019.01.27 - BG
2019.01.20 - TK
2019.01.19 - TK
2019.01.19 - BG
2019.01.17 - TK
This was a short test observation meant for checking
master calibration files, but it's somewhat useful for
excluding presence of dips.
2019.01.15 - TK
We are puzzled about the main dip being shallower in this
(TK) data than BG data.
2019.01.15 - BG
Note: There is no evidence of the "double-dip" feature that
was present on Jan 12.
2019.01.12 - TK
First observations with TK's new 15" telescope. No
calibration (dark or flat).
2019.01.12 - BG
Dew formed on my front corrector plate, starting (probably) ~
This dip is credible.
I'm skeptical about the two dips at the beginning.
This is the first observation of the season.
Imageand Basic Info
RA/DE = 11:48:33.6, +01:28:29. Observing season centered on
V-mag = 17.2, B-V = -0.08 +/- 0.04, spectral type = DBAZ
Distance from Earth = 142 parsecs (Izquierdo
et al., 2018)
WD mass =
0.63 ± 0.05 x Earth mass =
1.19e30 kg (Izquierdo et al., 2018)
WD radius =
1.32 ± 0.07 x Earth =
8419 km (Izquierdo et al., 2018)
WD Teff = 15,020 ±
520 K (Izquierdo et al., 2018)
A asteroid radius and mass = ~ 200 km and 1/10 Ceres,
i.e., ~ 1020 kg (Rappaport et al., 2016)
Kepler-based transit periodicities (A - F) = 4.49 to
4.86 hours, (Vanderburg et al., 2015)
Asteroid orbital speed (A - F) = 319 to 311 km/s
A asteroid time to cross WD diameter = 53 seconds (for
A asteroid transit depth ~ 0.56 mmag (assuming 200 km
WD1145 (blue circle) is at 11:48:33.59 +01:28:59.3 (J2000).
Red-circled stars have stability suitable for use as
reference. FOV = 27 x 18 'arc, northeast at upper left.
Analyses and Speculations
So far ground-based observations have detected dust clouds
with orbits associated with the K2 measurements of the A, D and
B periods, as shown below.
Figure M01. Scatter plot for some of the drift
lins of the past few years, converted to average orbit
Figure M02. Same data for just the inner-most
three orbits. The WD radius ~ 8400 km, so the A dust clouds
are typically ~ 700 km closer to the WD than their parent
Figure M03.Diagram, to scale, of the Kepler A, D and
B orbits, looking "down" from a "pole", including the 2015/16
dips associated with the Kepler A period, located at random
orbit asimuths but at their true orbit distances. The shape,
labeled "Most common dip size," corresponds to a dip size that
produces a dip depth of 30 % (which was comon for that
season). Since particles ejected with velocity components for
and aft are orbitally sheared to greater and smaller azimuth
values they define an oval shape that would keep expanding if
those particles didn't sublimate out of existence.
The above two graphs are based on plots like the following:
Figure M04.Waterfall plot for the 2017/18 observing
season, using the A ephemeris (as described in Rappaport,
plot for the current observing season, using the A
ephemeris (as described in Rappaport, 2018).
plot for the current observing season, using the D
ephemeris (described below)
This waterfall plot has evidence for the creation of several
fragments of the D asteroid on DOY 416 (2019 Feb 19).
The use of P = 4.552 hours is required by the 2019 Feb 12 light
Figure M06.This standard light curve shows two
"sharp" dips with identical shapes that are separated by 4.552
+/- 0.004 hours, which corresponds to the K2 D period (4.550
Figure M07.Detail of the first and second
appearance of a dip with identical structure on 2019 Feb 12(shown in previous figure) separated by 4.552 +/- 0.004
hours. The times of image exposures is shown by the x-location
of the red bars, and the average of a hypothetical source
function normalized flux during these exposure times is shown
by the red bar y-locations. The source function was adjusted
to achieve agreement with the observations.
This detail of the two dips shows that the 60-sec exposure times
had only a slight effect on inferred depth of the "source
function" (what would have been observed with short exposures).
The source function has a depth of 25 %. How can a dust cloud
that covers 25 % of the WD disk cross in only 2.5 minutes,
considering that a point source crosses the disk in only 0.9
minutes? This deserves an analysis of the geometry of cloud size
and crossing time.
Let's tell a story describing what may have happened in the D
orbit last week.
where Roche lobe surface has shrunk to coincide with the
asteroid's surface as the asteroid orbit shrinks, vs.
distance from WD1145, for a suite of asteroid densities.
The gravitational field at a distance of ~ 100 x WD radius is
strong enough to synchronize an asteroid's rotation with its
orbital motion. The surface of the Hill sphere (Roche sphere) is
where WD and asteroid D have equal gravity (in a rotating
coordinate system), and it will be approximately coincident with
the D asteroid's surface if the asteroid's density is ~ 3.2
[gm/cm^3], as shown in the above graph.
If the D asteroid has a density of ~ 3.2 [gm/cm^3]
it will be at risk of losing dust and fragments on its
surface when the smallest of ejection velocity outward is imparted
to the dust or fragment. In other words, there is no "escape
velocity" for such an asteroid, and the smallest nudge will send
surface material "floating" away from the asteroid. This
suggestion was made by Rappaport (2016) as a means for producing
a population of fragments in orbits interior to the asteroid.
The reason only interior orbits will be populated is that
fragments will more likely be ejected from the hot pole, the end
of the asteroid facing the WD, than from any other surface
location. (Note: the WD gravity field radial gradients are so
great at 100 WD radii that any asteroids at this distance will
be rotating synchronously with their orbital period.) The reason
for regarding the hot pole as the only source for fragments is
related to the fact that the hot pole will have a heat wave
penetrating the asteroid surface starting with a surface
temperature of ~ 1400 K, whereas the asteroid sides and opposite
pole will be much colder. When the heat wave encounters a layer
with volatile minerals the sublimation of solid to gas can
create a cavity with pressurized gas that will eventually
overcome the structural strength of overlying material and eject
the that material outward.
After this outward ejection of material, or fragment, a fresh
surface is exposed at two places: on the asteroid and one side
of the fragment. Both will initially be colder than the 1700 K
steady-state surface temperature, but they will begin to heat
and produce a "heat wave" that penetrates underlying material.
After the surface of each body reaches 1700 K they may "turn on"
the sublimation process at some level where the most volatile
mineral resides. The timing of when fragment dust clouds are
produced should depend on the rotation state of the fragment. If
it rotates fast the the steady-state surface temperature will be
lower than if it isn't rotating. Assuming the fragment achieves
synchronous rotation it will become hotter on the star-facing
end, and this is where dust production should occur. There are
several factors that can influence activity fragment level, such
as the time to achieve synchronous rotation, but also fragment
shape. It is well known that the bottom of moon craters are
hotter than flat surfaces, due to crater bottoms having a
smaller than hemisphere for radiating away heat. The same thing
can exist on the asteroid and its fragments. I have estimated
that 1900 K is possible, using the moon example. Crater bottoms
may therefore be favored locations for the ejection of dust, or
Here's a repeat of the
waterfall plot using the D ephemeris that was established by
the two dips analyzed in the previous two figures.
Repeat of Fig. M05.
In the above figure let's imagine that the D
asteroid has a large crater that has reached a surface
temperature of ~ 1900 K. Near surface sublimation occurred and
produced a small dust cloud at DOY 408. Since its source is the
asteroid it will have a center of activity fixed to the asteroid
period, not a period associated with an orbit at the asteroid's
front surface. The asteroid's front surface may be 100 km closer
to the WD than the asteroid center, This is indicated by the
first vertical green bar.
This activity subsided by DOY 413, shich accounts for a gap
between the vertical green bars. A subsurface heat wave reaches
lower levels that exceed the temperature of a layer with
volatile mineral, causing the formation of a gas chamber with
pressures that eventually exceed the strength of overlying
material. An explosive ejection of four fragments occurs on DOY
416. Since it was a crater on the star-facing surface that was
the source of the explosive ejection the ejected fragments will
be in elliptic orbits with a smaller average orbit radius than
the parent asteroid. None of the ejection velocity is lost by
overcoming the asteroid's gravity because the WD gravity field
matches the asteroid's gravity (in a rotating reference frame).
This is equivalent to stating that the Hill sphere radius (same
as Roche lobe) equals the asteroid radius. In theory, a rock on
the surface could "float away" with the smallest nudge.
After 4 or 5 days these fragments became active sublimators on
their fresh surfaces, producing the 3 or 4 drift lines sloping
off to the left.
The paper by Su, Jackson & 15 others (download) presents a
model that can account for observed variations of near-IR
thermal emission of a young star's debris disk. The model is
based on collisions that produce dust clouds with finite
lifetimes, and I think it has possible use in explaining WD1145
transit fades. Although the paper's purpose is to account for IR
thermal emission from the debris disk the same dust cloud model
could be used to explain transit dips of the star by such a disk
if it was inclined edge-on.
The WD1145 discovery paper (Vanderburg et al, 2015) suggested
that the transiting dust clouds are produced by the sublimation
of minerals on the surface of planetesimals (in the 5 Kepler
orbits). Presumably, a mineral molecule would sublimate from the
surface and combine with others to form a particle, and this
particle would continue to move away from the surface to join
other such particles and form a cloud. I personally preferred a
model in which a heat wave penetrated to a depth where a
volatile mineral was present, and when it sublimated the gas
pressure build-up until it overcame structural forces of
overlying material and explosively created a jet that ejected
the overlying particles. (From my moon work, 5 decades ago, I
knew that depressions were warmer than a surrounding flat
surface, and these depressions were favored sites for the
creation of "sublimation jets." Observations by the Rosetta
spacecraft, orbiting Comet 67P, found dramatic evidence for this
with deep pits being the source for ejected material.) Ever
since this model was under consideration there has been no
credible mechanism to account for the large ejection velocities
required for the particles in such clouds. Note that this model
assumes quasi-continuous production of dust due to either
surface sublimation or sublimation jets that eject overlying
dust. Whereas the discovery paper assumed the source of these
dust ejections was a planetesimal (on of the 5 associated with
periods A through F), later thinking favored dust coming from a
source of many fragments that drift away from the planetesimal
(due to Hill sphere shrinking to the planetesimal's surface).
Each dust particle would have an ejection location that its new
orbit would pass through. Although it is likely that particles
are ejected isotropically, with a range of speeds, it will be
useful to focus attention of the following 3 velocity
components: up/down, in/out and fore/aft.
Particles ejected vertically
(up/down) would be in inclined orbits. Their
maximum vertical extent would be determined by the ejection
speed; their orbital period would be unchanged. The particle
would pass through the fragment's orbit plane (and close to the
fragment) twice per orbit: once at the ejection location and
once at the opposite orbit location.
Particles ejected in a radial direction (in or out), either
toward the WD or away from it, would be in eccentric orbits
confined to the fragment's orbit plane; their orbital period
would also be unchanged. The maximum radial extent of the motion
of these particles would also depend on the ejection velocity.
These particles would return to their ejection location twice
per orbit: at the ejection location and the opposite orbit
Particles ejected parallel to the fragment's orbit motion,
either forward and
backward in direction (for/aft), would be in
eccentric orbits with different periods. Those ejected forward
would have their periastron at the ejection location and those
ejected backward would have their apoastron at the ejection
site. Because these fore/aft ejections lead to different orbit
periods these particles would undergo "orbit shear," also
referred to as "Keplerian shear." As time passes they would
spread out along the orbit in both directions.
Let's summarize the properties of the three ejection components:
1) up/down: Same P. Vertical range (above & below original
orbit plane) determined by ejection speed. All up/down particles
come together twice per orbit.
2) in/out: Same P. Distance
outward/inward from original orbit determined by ejection
speed. All in/out
particles come together twice per orbit.
3) fore/aft: Different P. Range of periods depends on
ejection speed range. Particles continue to spread out along
the orbit to distances proportion to time since ejection.
The ejection velocity vector for a typical particle
(assuming isotropic) will have components for each of the
three directions. The particle can therefore be expected to
return to the ejection location, and anti-ejection orbit
locations, every orbit, and during subsequent orbits slowly
spread out along the orbit either ahead or behind the source
fragment. Even particles belonging to the
first two categories will eventually move away from the source
fragment due to radiation pressure, collisions, charge
accumulation and interactions with other charged particles and
So far there is nothing exotic about the above scenario; the big
problem is that sublimation jets are thought to be incapable of
ejection velocities needed to account for the observed dust
cloud sizes. Dips are typically 4 minutes in duration, which
corresponds to a size that is ~ 5 times the WD disk diameter.
We've observed dip depths of > 60% on a few occasions. If the
dust clouds are opaque (with abrupt edges) then we can consider
them to have the shape of a band that's long in the orbit
direction, and narrow vertically, as it transits the WD disk. If
the obstructing dust cloud is a band that crosses disk center,
then to produce a 60 % blockage the vertical width of the dust
cloud would have to be 0.50 x WD diameter. This would require
that some particles are in orbits inclined
so that they extend to the + and - 0.50 WD
radius locations. For particles to reach those heights above the
source fragment's orbit plane they would have to be ejected with
a speed of 2 km/s (as derived below). So far no one has made a
case for sublimation jets being able to achieve these speeds.
Collisions, on the other hand, can achieve speeds of these
amounts [need a reference for this]. Whether a particle is
ejected by a sublimation jet or a collision, it will be subject
to the same celestial mechanics described above. A particle
ejected from a collision will have the same behaviors associated
with the up/down, in/out and fore/aft categories. The up/down
and in/out particles would be in orbits that return them to the
collision location, and anti-collision location, every orbit.
The fore/aft particles would initially join the other particles
for these convergences, but over time they would spread out
along the orbit. In other words, the dust cloud would expand and
contract twice per orbit! Before Keplerian shear produces
significant spreading out along the orbit it can be said that
all particles will fly apart and come back together "in phase" -
producing a dust cloud expands and contracts twice per orbit.
Note, however, that since the Earth's line-of-sight to the WD is
for just one orbit location we will observe the dust cloud when
it is at the same phase of its expansion/contraction cycle. It
will therefore produce the same dip depth for every passage in
front of the WD. None of our observations will reveal dust cloud
changes in size between transit events.
The interesting bonus for the collision model is that every time
the dust cloud contracts, the dust and any larger fragments of
the source fragment will pass each other at close range with
large relative velocities (on the order of 1 to 8 km/s). Ever
contraction therefore affords a new opportunity for additional
collisions! An initial collision may start a cascade of
subsequent collisions. This could be a source for replenishing
the dust cloud with new particles.
The cascade of subsequent collisions every half orbit could
replenish particles that sublimate out of existence during the
time it takes for them to spread along the orbit (due to
"Keplerian shear"). We don't know the timescale for particles to
sublimate out of existence, but if we knew the function for
particle ejection speed vs. particle size we could use the
observations of dip width to model the sublimation parameters.
For example, here's an observation of a dip that has a stable
width of ~ 0.04 phase units (11 minutes):
Figure M09.Dip width and
depth exhibiting stability for 3 months. This requires
stability of dust cloud size size along orbit direction
(width) and perpendicular to orbit direction (depth). Based on
a waterfall plot this dust cloud was created by a collision on
DOY ~ 71.
Here's the waterfall plot that exhibits dip drift lines (DLs)
that diverge from an apparent creation date of DOY ~ 71.
plot showing a set of drift lines (DLs) that
diverge from a specific phase location at DOY ~
71 and phase = 0.2. This DOY is presumably the
date of an initial collision that created four
clouds of dust and chunks of material that may
have continued to collide every half orbit for
the continued production of new particles and
smaller chunks. Properties for DL#1 are shown in
the previous figure.
DOY ~ 71 there was no evidence for the dust cloud
under consideration, which is additional evidence
that it was created by a collision on the DOY and
phase associated with the DL divergence location.
that for the original mechanism suggested for the production of
the dust clouds, sublimation jets, the dust cloud would remain
the same size throughout its orbit even though the particles are
constantly moving throughout the cloud (the only exception being
a continual spreading out along the orbit due to Keplerian
shear). For the collision model, however, particles and
fragments created by a collision will be in clouds that are
expanding and contracting twice per orbit, with coming together
events occurring twice per orbit, for possible subsequent
collisions and the creation of new particles.
Ejection Speed Requirement
What ejection speeds are required to account for the light curve
observations? The deepest dips are ~ 60 %. We don't know the
optical depth of the dust clouds, so consider the two extremes
of totally opaque and semi-transparent. A totally opaque dust
cloud that covers 60 % of the WD disk would produce the deepest
dip recorded. The next figure shows how much of a disk is
covered by an opaque band-shaped object with one edge
intersecting disk center and the other edge located at a
fraction of a radius toward a pole. For example, an opaque cloud
shaped like a band (with edges aligned with the orbit) that
extends from the equator to a latitude of 30 degrees will
obscure 60 % of a hemisphere.
fraction by an opaque cloud band (parallel
edges) with one edge intersecting the disk
center as a function of how far the other
edge extends toward a pole (blue trace). The
gree trace is for an opaque band "entering" from a pole
Consider a dust cloud that was opaque throughout the part
overlapping the WD disk (i.e., opaque to the "up/down" edges and
opaque from left limb to right limb). For it to produce a dip
depth of 60 % it would need a "vertical" extent of at least the
WD radius. This assumes the cloud center-line passes through
disk center; if the cloud band was offset from disk center such
that one edge coincided with a pole the other edge would be on
the other side of center at a projected latitude of 12 degrees
(which makes use of the green trace in the above figure). In
other words, a 60 % dip depth requires that cloud vertical
extent be at least 0.6 x R_wd.
There's one problem with this opaque cloud geometry? If high
opacity extended in the orbit direction beyond both limbs of the
disk, by even a small amount, the dip shape would be "flat
bottomed." We've never seen a "flat bottom" dip!
The optical depth alternative
is semi-transparency with a much larger cloud size. The
extreme example is a cloud whose transparency is uniform
across the WD disk. The deepest dip then requires that the
large cloud was 40 % transparent averaged across the entire
disk. We can't allow for a uniform 40 % transparency across
the disk for the same reason that we are uncomfortable with
the opaque cloud model: if transparency were
uniform across the disk, and extended just a small amount beyond
the limbs, a "flat bottom" shape would be produced. Therefore,
transparency for the deepest dip (and all observed dips) must
vary across the disk (i.e., along orbit direction). It would be
acceptable to assume that a dust cloud is opaque at the center
but the size of the opaque region must always be small in
relation to the disk.
The deepest dip depth of 60 %, combined with the lack of any
observed flat-bottomed dips, requires us to consider only dust
cloud models that are semi-transparent averaged over solid angle
regions (projected area regions) comparable to the WD disk. We
are also forced to explain dust clouds with vertical extents
greater than > 0.6 x R_wd. Cloud vertical extents of > 1.0
x R_wd (completely covering WD disk) is a reasonable
What ejection speeds are required for producing a cloud that
completely covers the WD? Consider the case of edge-on geometry
for the parent object (planetesimal, or asteroid) and the many
fragments associated with that parent object and which are the
source of dust clouds when they are active. The A-system of
fragments has a period of ~ 4.493 hours, corresponding to a
circular orbit speed of 319 km/s (assuming WD mass = 0.63 x
M_earth). For particle ejection speeds that are small compared
to orbital speed we can assume that up/down and in/out ejections
(i..e, those with no change in orbit period) will have
sinusoidal displacement variations with respect to the source
orbit. For example, a 1 km/s ejection directed upward will
correspond to an orbit that is inclined by 0.18 degree (57.3 x
1/319). The particle will undergo a sinusoidal displacement that
takes it above the original orbit plane a distance of 2600 km (1
x 4.493 x 3600 / (2 pi)) during its first quarter orbit, then to
the same distance below the orbit plane during the next half
orbit, etc. Note that 2600 km is 0.31 x R_wd (assuming R_wd =
8419 km = 1.32 x R_earth). The total vertical extent of its
travel will be 5200 km.
Imagine now that at the same time the first particle was ejected
upward at 1 km/s another particle was ejected downward at 1
km/s. The two particles would separate across a distance of 5200
km during the first quarter orbit, come back together during the
next quarter orbit, etc.. Particles ejected in the in/out
directions would be in eccentric orbits with the same plane as
the source orbit. These particles would have no vertical
displacements. If we add a uniform distribution of directions
for particles whose ejection directions are within a ring whose
axis is in the orbital direction (at the time of ejection) their
vertical displacements will form a (uniform) distribution of
vertically displaced locations that undergo two sinusoidal
expansion/contraction cycles per orbit. At the time of greatest
expansion this cloud of particles will extend 5200 km, or 31 %
of the WD's disk diameter.
Observers from different azimuthal directions would see the
cloud as it transits the WD to have a range of sizes, from 31 %
of the WD at two azimuths to just a tiny dot of ~ 0 % the size
of the WD disk at azimuths differing by 90 degrees. Each
observer would not know that the cloud changed size because they
would always observe it at the same phase of the cloud's
For those observers where a maximum size occurs at transit it is
possible to calculate that a 60 % dip depth requires ejection
speeds of at least 2 km/s. If the dust cloud is not opaque when
it has expanded to its maximum size the required speeds exceed 2
km/s. Full disk coverage for an edge-on orientation requires
speeds as high as 4 km/s. If viewing geometry is not edge-on,
even greater speeds are required!
For anyone who prefers sublimation jets instead of collisions as
the source of dust cloud particles the same speed requirements
Displacements in the radial direction for the in/out particles
cannot be calculated using the same concept employed for the
up/down vertical displacements. Consider the case of a particle
being ejected from a fragment in a circular orbit with P =
4.4930 hours traveling 319.063 km/s with an ejection speed in
the forward direction of 3.19 km/s (~ 1.00 % increase of orbital
velocity). The particle's radial line to the WD center of mass
will sweep out an area per unit time, dA/dt, that is 1 % greater
than the fragment. Circular orbits of the same central mass
exhibit the relation dA/dt ~ P^1/3 (based on ratio of semi-major
axis cubed being proportional to period squared). Or, P is
proportional to (dA/dt)^3. So for a 1 % increase in dA/dt the
new period should be 1.0303 times the original period. For the
case of ejection speed = 3.19 km/s (in the forward direction) P
should become 4.62914 hours, which is 0.136 hours later than the
fragment after one orbit. The fragment's orbit circumference of
612.9 x R_wd, so its speed is 136.4 x R_wd / hour. The later
arrival of the particle after one orbit corresponds to 18.55
R_wd, and this corresponds to a lag rate of 31 R_wd/day. Here's
a summary of ejection speed and cloud size in both the vertical
and radial directions for A-system dust clouds (assuming edge-on
viewing geometry and opaque clouds with abrupt edges):
km 0.03 R_wd 3
% 1 R_wd/day
km 0.06 R_wd 7
% 2 R_wd/day
0.5 km/s 1300
km 0.15 R_wd 20
% 5 R_wd/day
km 0.31 R_wd 40
% 10 R_wd/day
km 0.62 R_wd 72
% 20 R_wd/day
3.2 km/s 8,400
km 1.00 R_wd 100 % 31
5 km/s 13,000
km 1.55 R_wd 100 % 49
Here's an illustration, drawn to scale, of a dust cloud that can
produce a 30 % dip depth, if it is opaque (as viewed from
"above," looking down from a pole perspective):
to scale, of the Kepler A, D and B orbits, looking "down
from a pole." Symbols for dips observed during the 2015/16
season are shown at their correct orbital distances with
randomly chosen orbit azimuths. The shape, labeled "Most
common dip size," corresponds to a dip size that produces
a dip depth of 30 % (which was common for that season).
Since particles ejected with fore/aft velocity components
undergo Keplerian shear they will spread out along the
orbit as time progresses. The dust cloud will therefore
evolve from a spherical shape to an oval shape. If the
particles sublimate after a few orbits, and are
replenished by collisions occurring twice per orbit after
the initial collision, the oval shape can take on a
steady-state shape and size. A typical steady-state length
(along the orbit) is 5 x R_wd (half the width of this
Consider the suggestion that all dust clouds pass over the WD
disk center (zero impact parameter), they are semi-transparent
(with no completely opague regions), and they cover the entire
WD disk. Dip depth would then depend on the projected area of
particles covering the disk. A dust cloud that covers the entire
WD disk must have particles that were ejected at speeds as high
as 3.2 km/s. In one day the 3.2 km/s particles that were ejected
in the forward direction would extend behind the dust cloud
center by 31 R_wd. The particles directed in the opposite
direction at the same speed would be ahead of the source
fragment by 31 R_wd. The total length of the cloud would be 62
R_wd, which is 520,000 km. The fragment has an orbital speed of
319 km/s, so the entire cloud would require 27 minutes to pass
in front of the WD, or 0.10 phase units (that's ingress to
egress, not AHS tau1 + tau2). For Dip#1 in Fig. M09, the AHS
width is ~ 0.040 phase units. This corresponds to FWHM = 0.052
phase units, and 0.150 phase units for the 5 % level width. The
corresponding time intervals are AHS width = 11 minutes, FWHM =
14 minutes and 5% width = 40 minutes. The ratio of 40 minutes to
27 minutes leads to a duration of 1.5 days for the fastest
particles before they disappear.
The fastest particles are presumably the smallest ones produced
by a collision. Is it reasonable to require that 0.5 micron
radius particles disappear due to sublimation in 1.5 days? I
don't know, so this will be a question for modelers to address.
Data exchange files are
available in two formats: light curve details (one line per
image) and dip fits (asymmetric hypersecant (AHS) model fits
for each dip). The first of these is available for download
using the above links (though I recommend that anyone using
these data check with me for updates since I sometimes find
errors and post the corrected files here). Data exchange
files of the second format (AHS dip fits) may be requested
from B.Gary. These may also be available here in due time.
Please don't ask me to co-author a paper! At my age of
almost 80 I'm entitled to have fun and avoid work. Observing and
figuring things out is fun; writing papers is work.
My observations are now "in the public domain" for this
observing season (2018/19). If my data is essential to any
publication just mention this in the Acknowledgement section.
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Tom Kaye presentation at 2016 Society for
Astronomical Science meeting: link