KIC8462852 Hereford Arizona Observatory Photometry Observations #2
Bruce Gary, Last updated: 2017.09.10, 00 UT
I've decided to present to the public domain a web page that I've kept secret since Aug 28 (in order to avoid the notice of Reddit). It's for the "greater good" of science, I hope, that I at least give the serious investigators access to my observations. This 3rd web page for my KIC846 observations is at:  http://www.brucegary.net/ts3/ 


Figure 1.1. Light curve for "May to now" for HAO V-band observations, with an adjustment for an "Inverse Gaussian" model for the long-term variation/fade of OOT brightness (Fig's 1.5 - 1.8 show that model). Only observing sessions with at least 1.3 hours of observations above 30 degrees elevation and with mostly clear skies are included here. Notice the different asymmetries for the three major fade events: the May fade begins fast and recovers slower, the June fade begins slower and its recovery is interrupted by a small fade, while the August fade began slowly and recovered fast. I don't know if the last observation's departure from the OOT line is real; probably not.


Figure 1.2. Tabby Team r'-band data (as of Aug 07), graciously provided by Tabetha Boyajian to help in comparing with HAO V-band observations. I have adjusted the Tabby Team data for a hypothesized long-term fade model (the "inverted Gaussian" model, described below). The gray trace is my fit to the data using 7 AHS components (asymmetrical hyper-secant, used thousands of times for fitting dust cloud dips exhibited by WD 1145+017, link1 & link2). The AHS components #2 and #3 overlap, so their individual shapes are show with dotted blue traces. The 7 AHS components have been adjusted to provide a solution (minimum of sum-of-chi-squares) for both the LCO r'-band data and the HAO V-band data; the only exception is an additional free parameter for multiplying depth of Dip#2 for V-band. These r'-band data are from 16" telescopes at the Las Cumbres Observatory sites in Hawaii (OGG) and Tenerife (TFN).

The dip underway was at the ~ 0.7 % depth level on Aug 7 (in agreement with the HAO data, previous figure). I performed a sum-of-chi-squares minimization solution for all AHS parameters (4 parameters for eqach AHS component, or dust cloud). Comparing the above two figures it is obvious that the dip in June ("Celeste") has a depth ratio, D_r' / D_v, that differs from one. In fact, it is 0.24 (the Dip#2 r'-band depth is 1/4 of the V-band depth). This may be due to the mid-May dip ("Elsie") being devoid of particles smaller than 1 micron, while the early part of the mid-June dip ("Celeste") has an abundance of smaller than 1 micron particles. (Could this be explained by the Elsie dust cloud being closer to Tabby's Star than 0.2 a.u., with P = 30 days, whereas Celeste is farther away? Nope! Must have another explanation.)


Figure 1.3. HAO V-band normalized flux, fitted with AHS model that fits both the Tabby Team r'-band data and my HAO V-band data, combined, allowing for only one difference: the early part of the mid-Jun dip (Dip#2) has a depth that differs from the r'-band depth. It is 4 times deeper for the V-band data than the r'-band data. The August dip recovered faster than it began, whereas the May fade began fast and recovered slowly.

If the r'- and V-band data are accurate, then one could tentatively conclude that Dip#2 has a particle size distribution that includes small particles (smaller than 1 micron), whereas there is no evidence for the presence of this component of small particles for the other dips. Since I'm not that convinced of the quality of all of this data I won't adopt that tentative conclusion. So why did I go to the trouble of performing this tedious analysis? To illustrate what I think should be done when better data is available. (For example, maybe the Thatcher Observatory V-band data is better than mine, or maybe some professional astronomer has used a big aperture telescope to monitor in two bands, and we don't know about those observations yet - because that's what professional astronomers like to do.) By the way, the AHS analysis for combined data sets at different wavelengths has been done for WD 1145+017 in an article that has just been submitted to MNRAS, and will soon be posted at arXiv (Siyi et al). I'll give a link to the arXiv article when its available.
How important is it to correct for long-term fade? There are three reasons for this: 1) We don't know what's causing the long-term fade (more accurately described as variations, just a fade recently), so it's premature to assume that the cause is dust, 2) even if the long-term variations are caused by dust they are several "orders of magnitude" different in timescale (years or decades vs. a few days) that their properties could differ greatly, and therefore using the same parameters to describe them, and solving for values, could be very misleading, and 3) whatever the explanation for the long-term fade/variation when a dip occurs we want to know about that specific dip's properties (i.e., depth vs. wavelength dependence, as produced by the particle size distribution of the particles in the dust cloud causing the dip). Keep in mind that the current fade rate is well-established (by 3 observers), so adjusting for it does not strain objectivity. The next figure is what Tabby sent me, which is uncorrected for long-term fade (the long term variation is currently a "fade").

  
Figure 1.4. This is the Tabby Team r'-band data provided to me by Tabby, on Aug 07. An "inverted Gaussian" long-term fade model is superimposed (gray trace), since this data doesn't allow for what I believe is a real OOT long-term fade (shown in Fig's 1.4 - 1.6).   


Figure 1.5.  These are my V-band observations from the past year, plus some "clear filter converted to V-mag" observations. Dip data are shown with open circles. An "inverse Gaussian"  long-term variation (fade) model is shown (gray trace). Superimposed on the inverse Gaussian model is a sinusoidal model that is marginally statistically significant (at the ~ 5-sigma level). The sinusoidal period is 34 days and peak-to-peak amplitude is 1.2 mmag.
 

Figure 1.6. A 2-year span of observations showing the need for a non-linear long-term variation (fade) model for matching measurements that are judged to be "out-of-transit" (OOT). Data taken during dips are represented by open circle symbols. The gray trace is an "inverse Gaussian" model.(The data within 300 < DOY_2106 < 360 exhibit a 29-day sinusoidal variation with semi-amplitude = 2.9 mmag. With only 2 cycles of coverage I hesitate to claim that such a variation is real; more observations during an OOT state are needed to confirm this possibly interesting finding.)

Is there independent evidence for the long-term fade that I'm proposing? Yes, from two other sources (the 2nd source is ASAS-SN observations, described in the next section). The first data source confirming the HAO fade during the past 1.5 years is from AAVSO data submissions by David Lane (LDJ), shown in the next figure.


Figure 1.7. David Lane's V-mag measurements (red circles) are superimposed on the previous graph. Dip data have been omitted. An offset of 0.07 mag was applied to the LDL data to achieve empirical agreement (related to use of different reference stars with different catalog V-mags). The LDL data has been median combined in groups of 9. 

The David Lane V-band observations are not only compatible with the HAO V-band measurements but they argue for the same long-term fade rate for the interval where our two data sets overlap.  

Note: If we adopt the 2015 magnitude as the 100 % level for normalized flux, then the latest dip would be at a normalized flux level of 97.5 % (i.e., depth of 2.6 %). I prefer to state that the dip is ~ 1.0 % below the long-term fade level. As a future publication will show, the non-dip brightness during the past decade is too variable to use for establishing a reliable 100 % level; it just varies too much! An additional reason for "correcting" a several-month long LC for a best-estimate long-term fade model is that when this is done it is possible to fit the dip structure using the AHS model, as demonstrated in Fig's 1.2 and 1.3. Using AHS permits other physical phenomena to be studied, such as which dips exhibit unusual depth ratios (e.g., D_r' / D_v), which in turn can be used for the interpretation of "particle size distribution" differences between the different dust cloud dips. That's why I suggest describing the dips as departures from the long-term fade variations. ("That's my story, and I'm sticking to it!")


Figure 1.8. This 4-year light curve, with almost 2 years of data, is fitted with the "inverted Gaussian" model.

The "inverted Gaussian" model is compatible with the notion that brightness could level off next year (2018) and begin to recover the following year (2019), eventually returning to normal. However, this is just an extrapolation of a mathematical model, not based on a physical model, so extrapolating is really unjustified. However, this behavior is what a dust cloud model for explaining the long-term fade could predict, as described by a plausible physical model described below, link. Of course, the model could only be an over-simplification of the real situation, but it's a "guide" for what might happen.

ASAS Observations of Long-Term Variability

Joshua Simon et al. have published (link) 11 years of ASAS V-band observations, and 2 years of ASAS-SN V-band observations (both using small ground-based telescopes). These are the best quality long-term monitoring observations (longer than the Kepler 4-year data set) so far of Tabby's Star.
 
The two V-mag series agree where they overlap, and the ASAS agrees with the 4-year Montet & Simon Kepler fade pattern. What's more interesting, though, is year timescale brightenings in 2006/2007 and 2014. The latter was 4 %. During the last 2 years there's a slow fade, consistent with my fade over the same time. With this perspective I suggest that we can reinterpret the century of DASCH data in the following way: decade timescale variations with a peak-to-peak amplitude of 4 or 5 % could have been occurring during the past century, but the DASCH data were simply too noisy to have shown such variations. (The abrupt 0.12 mag shift in brightness, occurring across the infamous "Menzel gap," is related to the change of plate emulsion and telescope change that occurred at that time ~ 1960); therefore, we are justified in discounting the suggestion (Schaefer, 2016) of a century timescale "fade."

Simon et al. suggest that the 8-year interval between brightening events of KIC846 might be evidence for an 8-year variability due to a magnetic activity cycle. They keep open the notion if interstellar medium structure since their I-band LC shows long-term changes that are the same (within the noise) as those for V-band; they recommend that future monitoring projects include two bands to investigate wavelength dependence of long-term brightness variations.

The most conservative stance to take, I suggest, is that variations of as much as 5% occur on decade timescales. (Or, just because a 2-year snippet shows a fade doesn't mean that Tabby's Star will fade into oblivion, any more than a 2-year snippet showing a brightening would mean that Tabby's Star will some day outshine our sun.)


Figure 1.9.  ASAS V-band observations during the past 11 years (top panel) with the Montet & Simon (2016) Kepler data overlain (lower panel, red) and ASAS-SN data overlain (bottom panel, blue). This figure was taken from Simon et al. (2017). 

Links Below 
    11-years of ASAS V-band monitoring 
    Why this web site 
    Basic Infor for KIC846
    Can Structure Within an Observing Session be Believed?  
    List of Observations
    Daily Observing Session Details
    Tutorial on Fading Star Basics
    Comparing Kepler Data with Recent Data
    Long Term Trends
    Plausible Physical Model for Everything 
    A Prediction  
    References  

    Earlier Web Page (#1)  started 1.7 years ago, and no longer updated


Why This Web Site?

This web page was started Jun 18 with the original intention of avoiding the attention my observations were getting at "reddit." I knew nothing about reddit until a couple people e-mailed me in mid-June with incidental mention that some people were criticizing my observations. When I checked the reddit "thread" for Tabby's Star I was initially unimpressed with the quality of some of the postings, so that's when I decided to discontinue daily updates of my KIC846 web page (http://www.brucegary.net/KIC846/). I intended this web page to be for my personal use. However, after more than a dozen followers of the first web page wrote to express their appreciation for my observations, with a hope that I would change my mind about discontinuing updates, I decided to give them the URL to this web page. A few days ago someone pointed out that the reddit thread was making frequent use of my observations I took another look at the reddit commentary, and was quite surprised to see frequent reference to my observations. I also noted that some postings were quite well-informed, and reasonable. I have now decided to open this web page to the "public domain." I'm aware that observations on this page will somehow show up at reddit, and I am now OK with this.

By the way, I'm not a professional astronomer. I'm an amateur, and I don't have any special background for having an informed opinion about what is causing the Tabby Star fades. My only qualification for contributing to the Tabby Star mystery is experience in performing quality photometry.

Basic Info for KIC846

RA/DE = 20:06:15.5, +44:27:25
All-sky photometry: B = 12.493 ± 0.025, V = 11.912 ± 0.025, B-V = +0.581 ± 0.035, as measured by B. Gary in 2016, link.
APASS Mag's: B = 12.360, V = 11.852 (B-V = +0.51), g' = 12.046, r' = 11.697, i' = 11.554
There's a 0.11 mag discrepancy between BV mag's in Boyajian et al (Table 3) and APASS (article is brighter).
There's a 0.23 & 0.21 mag discrepancy between B & V mag's in Boyajian et al (Table 3) and my all-sky V-mag (which are fainter than Boyagian et al)
Distance (based on Boyagian et al V-mag) is 1480 light years (454 pc).
Star Teff = 6750 K. Star radius = 1.58 x Rsun
Rotation period = 21.11 hrs (but should vary with latitude).
Brown dwarf (?) star 1.96 "arc (900 a.u.) to east. Too far to be gravitationally important now (but we don't know its past path)
Question: How do my HAO observations compare with the Tabby Team Las Cumbres Observatory observations? Consider the following two figures.

3. Can Structure Within an Observing Session be Believed? 

Whereas the process of using reference stars to provide a magnitude calibration for an observing session may be subject to an uncertainty of 3 mmag (0.3%), for example, all observations within the observing session may share the same systematic error and therefore provide other information, such as slope within the observing session. If such a slope within an observing session can be trusted then we would have additional information that could be used to create an improved model for what was happening between observing sessions. I have attempted to make use of this slope information in creating the following figure. 


Figure 3.1.  This 2-month light curve is modeled in a way that makes use of information about the slope of brightness within observing sessions.

At this time I do not know whether to trust slopes in this way. It should be kept in mind that the overall level of each observing sessions measurements is expected to be uncertain at an estimated 0.3 % level for good observing conditions, and up to ~ 0.4 % for poor observing conditions - as the following analysis shows.  


Figure 3.2 "Good" and "bad" observing sessions exhibit different uncertainties vs. reference star brightness. A "good" observing session has at least 2 hours of clear weather observations above 30 degrees elevation, while "bad" observing sessions have less than 2 hours of good observing (some have clouds throughout the session). Among the 40 observing sessions in this analysis 30 are "good" and 10 are "bad." The model fit is for the "good" observing sessions and it has two components: 1) "stochastic uncertainty," that depends on signal-to-noise level, and 2) "systematic error," which is a fixed value (i.e., the same for all stars). The model is a sum of both components. Since KIC846 has V-mag ~ 11.9 it is predicted to exhibit total uncertainty (stochastic plus systematic) of ~ 2.2 mmag (or 0.22 %).

At the present time I don't know how much to believe about the normalized flux trends within an observing session, when uncertainties should be mostly "stochastic" (with systematics shared for all data). Here's an example of internal consistency at the 1.0 mmag level. Since it is for one of my best observing sessions I think it can serve as a lower limit on level of variations that can be searched for as possibly real: namely, 1.0 mmag or 0.1 % of normalized flux.


Figure 3.3. Internal consistency during an observing session, suggesting that variations of 14-minute averaged data exhibit an RMS uncertainty of 1.0 mmag (0.1 % of normalized flux). 

If this argument is valid, then the following light curve segments (showing 1-hour averages, with an expected SE of no better than 0.5 mmag, or 0.05 %) may be worth considering.


Figure 3.4. Expanded date scales for two date regions, showing possible model fits that involve short timescale variations. It's too early for me to be "a believer."  


List of Observations (for all earlier observations, before Jun 18, go to link)


2017.08.28 V  
2017.08.25 V  
2017.08.23 V  
2017.08.21 V  
2017.08.17 V  
2017.08.16 V  
2017.08.15 V  
2017.08.14 V  
2017.08.12 V  
2017.08.09 V  
2017.08.08 V  
2017.08.07 V  
2017.08.06 V  
2017.08.05 V  
2017.08.04 V 
2017.08.03 V  
2017.07.31 V  
2017.07.09 V  
2017.07.08 V  
2017.07.07 V  
2017.07.06 V  
2017.07.02 V
2017.07.01 V  
2017.06.30 V  
2017.06.29 V
2017.06.28 V 
2017.06.27 V 
2017.06.24 V  
2017.06.23 V  
2017.06.21 V  
2017.06.20 V  
2017.06.19 V 
2017.06.18 V 


Daily Observing Session Information (most recent at top)


2017.08.28, V-band, 7.5 hrs of useful data   DataExchangeFile    


It's a little concerning that as noise level rises, which is probably due to air mass increasing, the brightness also increases.


I didn't use data after10.0 UT, when air mass rose above 2.0, because I didn't want noisy data to "contaminate" better quality data.


The bluest reference star is consistently "below" the model fit trace. This just means that it could use a fine adjustment of ~ 5 mmag for its adopted V-mag. I won't do it because then I'd have to reprocess all previous LC sessions.

2017.08.25, V-band, 8 useful hrs, DataExchangeFile  







2017.08.23, V-band, 2.7 hrs, DataExchangeFile  







2017.08.21, V-band, 2.5 useful hrs,  DataExchangeFile  







2017.08.17, V-band, 2.1 hrs   DataExchangeFile  







2017.08.16, V-band, 2.1 hrs, DataExchangeFile  







2017.08.15, V-band, 2.5 hrs useful data, DataExchangeFile  







2017.08.14, V-band, ~ 3 hrs  DataExchangeFile







2017.08.12, V-band, ~ 5.0 useful hrs,  DataExchangeFile







2017.08.09, V-band, 3.9 hrs, DataExchangeFile  







2017.08.08, V-band, 3.5 useful hrs  DataExchangeFile  







2017.08.07, V-band, 2.0 hrs of useful data    DataExchangeFile   

I had to "dry out" the inside of my (closed tube) telescope twice to evaporate water condensation on the corrector plate. This problem will eventually go away, but only when ambient water vapor dew point goes significantly below its current 60 F.







2017.08.06, V-band, ~ 1.32 useful hrs,  DataExchangeFile  

Although the sky was clear, the high humidity during the past few weeks raised the amount of water vapor inside my telescope tube so much that when the optical surfaces cooled tonight they frosted over with dew. I used a hair dryer to evaporate dew from the corrector plate but then the water vapor condensed on the primary mirror! 







2017.08.05, V-band, 2.4 hrs, DataExchangeFile  







2017.08.04, V-band, ~5.0 hrs, DataExchangeFile  

The observing session was plagued by clouds, but during clearings there was a consistent normalized flux of ~ 0.3 % below OOT level.






This calibration looks OK, i.e., unaffected by intermittent cloudiness.

2017.08.03, V-band, 1.3 hrs DataExchangeFile  

I was "desperate" to observe, so when a "sucker hole" opened up, I opened my observatory. Take this observing session result "with a grain of salt."
It would be my opinion that a new dip is not underway.







2017.07.31, V-band, 2.8 hrs  DataExchangeFile  

Either KIC846 still hasn't recovered, or my long-term fade model is under-estimating the fade acceleration!

 





2017.07.09, V-band, 4.4 hrs (of usable data)  DataExchangeFile  




Clody until midnight, then partly cloudy.



2017.07.08, V-band, 8.1 hr    DataExchangeFile     







2017.07.07, V-band, 8.0 hr DataExchangeFile  


I don't know if the downward slope is real.





2017.07.06, V-band, 0.7 hr  

I was desperate for a measurement, so I observed just before a thunderstorm arrived.





2017.07.02, V-band, 8.1 hrs,  DataExchangeFile 







2017.07.01, V-band, 8.0 hrs  DataExchangefile   


Averages of 14 minutes exhibit an RMS scatter of 1.0 mmag. (I want to thank Rafik Bourne, or Perth, Australia, for studying my "data exchange file" magnitudes and prompting me to identify outliers and rely more on better quality observing sessions. This graph, showing millimag scatter, was one of his suggestions.) 


The two processing methods disagree by 0.3%. I'm concerned about this and will continue to investigate.





 

2017.06.30, V-band, 8.1 hr  DataExchangeFile  


During 1.2 hours I took 10-sec exposures, unfiltered, which show a 52-minute periodicity with a semi-amplitude of 4.6 mmag.

This short-term variation is similar to what I measured 1.7 years ago; I measured a 49-minute variation with 1.8 mmag semi-amplitude: link 


New data product: differential photometry using 7 "same color" reference stars.

 





2017.06.29, V-band, 7.8 hrs  DataExchangeFile  







2017.06.28, V-band, 6.8 hrs DataExchangeFile







2017.06.27, V-band, 3.6 hrs, DataExchangeFile  







2017.06.24, V-band, 2.2 hrs, DataExchangeFile  




Thick clouds ended observing session.




2017.06.23, V-band, 1.7 hrs, DataExchangeFile  

Cloudy the entire observing session (up to 2.0 mag of loss), so data quality is poor.







2017.06.21, V-band, 5.4 hrs DataExchangeFile

Skies cleared at midnight, so I got ~ 5 hrs of acceptable images.







2017.06.20, V-band, 3.6 hrs DataExchangeFile  







2017.06.19, V-band, 8.1 hrs, DataExchangeFile  







2017.06.18, V-band, 5.1 hrs, DataExchangeFile 


Recovery trend is too slow to be apparent in this 5-hour observing session.


Good observing conditions (no wind, moon far away & faint, seeing OK).


Good calibration, so overall V_mag will be accurate.


Tutorial on Fading Star Basics (Reasonable Assumptions and Some Common Misunderstandings)

Dust Cloud Appeal

Modelers like dust clouds when trying to explain star fadings that aren't obviously due to binary eclipses or exoplanet transits. This is because for a given amount of mass small particles can block more light than a body with the same mass. For example, if a 10 km diameter asteroid is pulverized to 1 micron radius dust particles it is possible for such a cloud to obscure (scatter and absorb) 20 % of Tabby Star's light, whereas the original asteroid would obscure only 0.00000000002 %. This dramatic difference is due to the fact that the ratio of area cross-section to volume (i.e., mass) is proportional to 1/size.

Dust Cloud "Optical Thickness"

Dust clouds are almost certainly causing the fades of Tabby's Star, hereafter abbreviated as TS. That's my position, and I acknowledge that some reasonable people will want to emphasize an alternative. If the particles are dispersed enough that most of them don't overlap along a typical line-of-sight, we say that the cloud has a small optical depth. This means that some (possibly most) starlight will penetrate the cloud when the cloud passes in front of the star. An optically thin cloud can cover the star completely without producing a large fade. As an alternative, a dust cloud can be "tightly packed" such that essentially no star light passes through. That cloud is referred to as "optically thick." If such a cloud has a small projected area ("solid angle"), which is much smaller than the star, it could produce an equally small amount of fade as it passes in front of the star. Both cloud types can produce a 1% fade, or a 20% fade, or any fade amount. It is observationally difficult to distinguish between the two cloud types (optically thin vs. optically thick). Observations that show a smaller fade amount at longer wavelengths (e.g., smaller fade at red vs. blue) would constitute good evidence for an optically thin cloud with particles having radii of about a micron. However, if the same size particles are in an optically thick cloud, with sharp boundaries, then it would produce fade amounts that are the same at all wavelengths. Large, solid objects will produce the same fade amount at all wavelengths.

When Kepler measured a dip with depth 20% it could have been produced by an optically thick dust cloud that covered 20% of the star's solid angle (projected area), or it could have been produced by an optically thin cloud that covered the entire star but had an optical depth of only ~ 0.22 (noting that e-0.22 = 0.80).

IR Excess

The temperature of the dust will depend on its proximity to the star. If, for example, it's at 1.6 a.u. from TS (P = 618 days), the dust will be at a temperature of ~ 450 K (+180 C). Or, consider a dust cloud in a closer orbit, at 0.52 a.u. (P = 114 days); dust could be as hot as ~ 800K. Imagine that the dust cloud is optically thick (at Near IR wavelengths) and that it has a projected area 100 times greater than the star's projected area. The next graph shows a predicted "spectral energy distribution" for the star, the dust cloud, and star plus dust cloud.


Figure 2.1. Spectral Energy Distribution (SED) of a hypothetical star and large (optically thick) dust cloud. The red diamonds are from measured magnitudes of TS, the matching SED is for Teff = 6400 K (blue trace), and a dust cloud spectrum is shown for Teff = 800 K and projected area 100 x TS (brown trace). The gray trace is the sum of both components, and corresponds to what should be measured for such a hypothetical system.

This graph shows that for a dust cloud to show up in measurements at IR wavelengths (i.e., called "IR excess") it would have to: 1) be much larger in projected area than TS, 2) be optically thick, and 3) be hot (close to TS, orbiting with a short period). An optically thin cloud would have to be either hotter (in orbit closer to TS), or larger than the hypothetical model shown in the figure, for it to produce a meaasureable IR excess. It is therefore possible for extensive fadings to occur without an IR excess to be measured. In other words, the lack of a measured IR excess does not rule-out the presence of extensive dust clouds.

IR Excess Flux Nonsense: Some comments at "reddit" (which I rarely view and never contribute to) claim that if the long-term fade is increasing (and possibly accelerating), there should be an increase in IR flux. Not true. As described in my discussion of IR excess below a dust cloud that's orbiting at a distance of 0.25 a.u. (with a period of  40 days), will have a temperature of 800 K, and if it is opaque and covers a solid angle 100 times Tabby's Star, it will produce an IR excess that is too small for detection (as shown by a SED graph, below). If you relax any of these assumptions (e.g., more distant, optically thin, < 100 x solid angle of TS), which the present fade data permit, the IR excess will be even less detectable. Therefore, the lack of any observed increase in IR flux (at wavelengths < 5 micron) does not argue against the presence of a dust cloud that is being considered for explaining the observed long-term, slow fade.

Distance/Period Relationship

Given that TS has a mass estimated to be 1.43 times solar, we can relate planetary orbital radius and period using the following:

    a3 = 1.43 × P2

Where "a" is circular orbital radius in astronomical units, a.u., and P is period in years. This relationship is shown in the next graph.


Figure 2.2. Orbit radius vs. period (blue line). The apparent radius of TS (degrees) as viewed by an object in an orbit with the x-axis period is also shown (green dotted line).

Here's an example of how to use the above graph. Suppose we are considering Kepler dips #5 and #10 to be produced by the same object. They are separated by ~ 756 days. An object orbiting with that period would be at 1.9 a.u., and from that orbital distance TS would appear to have a radius of 0.23 degree. If this object's orbit were inclined with i = 89.77 degree, it would produce a grazing transit as viewed from Earth.

Temperature of Asteroids and Dust vs. Orbital Distance

An object without atmosphere, such as an asteroid, will be heated by TS to temperatures that depend on whether the object is rotating slowly (such as synchronously) with a period that is much shorter than orbital period. For the synchronous case, and assuming a Bond albedo of 10% (fraction of star radiation that is reflected), the sub-solar tempearature is given by

   Tsurf (ø) = (579 / sqrt(r) ) * (cos (ø) )1/4

where  ø is starlight incidence angle (zero degrees at sub-solar point) and r = asteroid/TS distance [a.u.]. The following graph was calculated from this equation.


Figure 2.3. Temperature of an airless planetesimal (asteroid) vs. orbital period (circular orbit). Red trace is sub-stellar location of a slow-rotating asteroid; blue trace is an estimate for temperature along equator of a fast rotating asteroid. Dust could be at temperatures ranging from the blue dotted trace to above the red solid line trace (as explained in text).  

Using the same example as in the previous section, with P = 756 days, a slow-rotating asteroid will have a sub-stellar temperature of ~ 430 K. A fast-rotating asteroid will be somewhat cooler, at ~ 300 K (23 C, or room temperature).

Dust is more complicated, since when a particle is small compared with the wavelength of thermal emission (a matter of microns for room temperature dust) the dust has a low emissivity, and it becomes hotter than nearby large particles. A calculation has recently been made by Rappaport and van Lieshout (private communication, 2017), and one aspect of their calculations is shown in the following graph.


Figure 2.4. Temperature of dust particles for large radii (> 10 micron, solid red line) and small radii (< 0.5 micron, dashed red line) versus (circular) orbital distance from the KIC846 star. Orbital period is shown (green dashed line). [Based on unpublished work by Rappaport and van Lieshout, 2017.]

It should be noted that in the above graph there would be no need to consider the "small particle" temperature result for distances smaller than about 0.2 a.u., because small particles at that distance from KIC846 would be so hot that they would immediately sublimate and disappear. Even the large dust particles would disappear at distances closer than ~ 0.07 a.u..

Given that Tabby's Team has found evidence for a wavelength dependence of depth (deeper at short wavelengths) we can assume that a component of small particles exist. How small? Smaller than ~ 1 micron. Since these particles will be among the hotter than larger particles, we may conclude that the dust clouds that produced the May and June fade features are in orbits farther from the KIC846 star than ~ 0.2 a.u.. These distances have corresponding orbit periods of 30 days and longer. Therefore, any suggestions that the May and June fades will repeat at intervals shorter than ~ a month will have to confront the finding of depth dependence upon wavelength.

Importance of Inclination

Inclination of the orbits of the dust clouds is a very important parameter for deriving a physical model to account for observations. Inclination was for some silly reason defined to be 90 degrees for an edge-on orbit orientation, and zero degrees when the orbit is viewed "face-on." An exoplanet won't produce transits for an inclination of zero degrees, and will always produce transits when inclination is 90 degrees. For exoplanet transits in which the entire projected area of the planet passes within the disk of the star will produce a fractional loss of starlight given by (Rp/Rs)^2, where Rp is planet radius and Rs is star radius. An observer outside our solar system, and located in the ecliptic plane, would see Jupiter transit with depth of 0.4% (0.004 magnitude, or 4 mmag). If the planet, or asteroid, orbits in front of the star in a way that appears to pass through the star's central point, we say that the transit path has an "impact parameter" of zero. If it passes tangent to the star's edge, we say the transit path has an impact parameter of one. Dust clouds produced by a solid object, such as an asteroid, will start out with the same orbit as the object, so such a dust cloud's inclination will be the same as the object's. The concept of "impact parameter" for a dust cloud loses meaning the larger it gets. When small, such a cloud's impact parameter will have to be < 1 in order to produce a fade, but if the cloud becomes large it can produce a fade even for impact parameters > 1. Tabby's Star is suggested to have a radius 1.58 x Solar Radius, so Rs = 1.1e6 km for Tabby's Star. An object orbiting at 1.6 a.u., for example (i.e., 2.4e8 km), will appear to have a radius of 0.26 degree (same as our sun as viewed from Earth). For transits of objects at 1.6 a.u. from Tabby's Star to produce transits the inclination of the object's orbit must be closer to 90 degrees than 89.75 degree (i.e., i > 89.75). Only one out of 350 star solar systems will have such a favorable orientation (producing transits of exoplanets at such an orbital distance).

A Possible Viewing Geometry

Imagine a situation in which an object is orbiting with a radial distance from TS of 2.9 A.U.; it will have a period of 1512 days (notice that this is twice 756 days). The star will have an apparent radius of ~ 0.3 degree as viewed by the object. If the orbit is inclined 89.7 degree, as viewed from Earth, the object will appear to pass in front of the star close to an edge of the star (one of the poles). The next figure is a diagram showing dust clouds passing over the north pole of TS, moving to the right. Only about half of the cloud will block starlight during a passage.


Figure 2.4. As viewed from Earth the projected path of dust clouds (3 tan blobs) in an orbit inclined 89.7 degree will pass in front of the "north pole" of TS on the near side of the star, moving to the right. Nothing is observed when the same dust clouds orbit on the far side, behind the star (lower path, moving to left). This geometry is for an orbit distance from TS = 2.9 a.u., with a period of 1512 days. The dark dot corresponds to a planet with a radius of 30,000 km, which is slightly larger than Neptune (or 0.4 x R_Jupiter), and it produces a dip depth of 0.08 % (0.8 mmag). [This diagram is borrowed from a modeling exercise for WD1145, which has many dust clouds orbiting with a period of 4.5 hours. A ring system is shown that accounts for an IR excess. The dotted region inside the ring system depicts circumstellar gas that extends in to within 10 radii of the star. The ring system and circumstellar gas disk are not present for TS.]

Dip Depth and Planet/Object Size

Figure 2.4 shows a planet slightly larger than Neptune (1.2 x radius) as a black dot in relation to the disk of TS (1,100,000 km radius). Such an object would produce a dip depth of 0.08 % (or 0.8 mmag). Since dip depth is proportional to (R_planet/R_star)2, to achieve a depth of 1%, for example, we require R_planet = 0.1 x R_star, which is 110,000 km radius (or 1.5 x R_Jupiter). The following graph shows dip depth vs. planet radius.


Figure 2.5. Dip depth vs. planet radius (or, optically thick dust cloud with equivalent circular radius).

The reently measured dip depths of 2% require either a planet size of 150,000 km radius (2.2 x R_Jupiter), or an optically thick dust cloud with the same equivalent circular radius. The same depth could be achieved by a dust cloud that covers the entire star that has an average optical depth of ~ 0.02.

A Way to Dispell Mega-Structure Speculation

Suppose it is determined that dip depth varies with wavelength. Dust clouds would provide a ready explanation for such a finding. In addition, the particle size distribution could be modeled since the wavelength where fade depth changes occur would indicate the wavelength where the transition between Mie scattering and Rayleigh scattering occurs, which in turn would indicate the typical circumference of the dust.

Mineralogy of the dust particles might also be subject to constraint, since scattering and absorption, as specified by a mineral's real and imaginary parts of the dielectric constant, influence the shape of extinction (scattering plus absorption) vs. wavelength. Such a modeling effort would require observations of high quality. The resulting dust cloud model would correspond to low optical depth with large areal extent.

A structure made of metal would not block light differently at different wavelengths (assuming the metal is thicker than the dimension of a few molecules). Therefore, if it is found that dip depth is less at longer wavelengths (i.e., red and infrared) vs. shorter ones (blue or green), then this would present a serious challenge to those who want TS fades to be caused by alien structures.

A Plausible Model for Accounting for Brief Fades and an Inverse Gaussian Model for Long-Term Fading and Recovery
Short Version: The long-term fade may be produced by a stretched-out dust cloud in an outer orbit whereas the brief but deeper dips that have produced the most excitement are produced by small dust clouds in one or more inner orbits. The stretching-out of the outer orbit dust cloud is straightforward, since any expansion of particles (from a collision) would send some particles in slightly different orbits, with slightly different periods. If the periods have a range of 1%, for example, after 100 orbits the dust cloud would have a torus shape, and it would be capable of obstructing starlight continuously. As the torus circular cross-section expands (without changing orbit size), one edge of the torus would eventually enter our line-of-sight to the star. This is when we would observe the beginning of a gradually increasing fade. The torus may expand to completely obstruct the star, but when that happens it could be optically thin and produce only a small fade. Continued expansion could be accompanied by diminished loss of dust density along our line-of-sight, so expansion should eventually be associated with a recovery of star brightness. The “inverse Gaussian” model that I have employed for fitting the out-of-transit observations calls for a maximum fade amount of ~ 3 % in one or two years, followed by a slow recovery to normal star brightness in 5 or 10 years. During all of that time there will probably be short fades, lasting a few days, produced by dust clouds in orbits much closer to the star (but not closer than ~ 0.2 a.u.), with orbit periods of at least 4 weeks (but not less, due to the temperatures for closer orbits causing the particles to sublimate to gas).
Assume that the KIC846 system has recently undergone collisions, possibly related to a passing star disrupting the "Oort" cloud of planetesimals (objects of all sizes), some of which would have changed orbit eccentricity causing them to enter the inner solar system. Suppose further that collisions occurred with existing planets in the inner solar system, orbiting at 0.2 and 2.0 a.u.. The resulting debris would produce clouds orbiting with periods of 30 days and 900 days. The temperature of the dust will range from 750 K to 1300 K (for the inner orbit) and 300 to 500 K (for the outer orbit). [These temperature calculations are based on unpublished work by S. Rappaport and R. van Lieshout, private communication, and soon to be submitted for publication in MNRAS by S. Xu et al. As an aside, these calculations can be viewed as ruling out smaller orbits, with shorter periods, for causing the brief fade events first discovered by the Kepler K2 mission, and recently documented and given names Elsie and Celeste. I won't explain more until the aforementioned articles are in the public domain.]

The planets are in orbits that are viewed almost edge-on, inclined to our line-of-sight by a small angle, such as 1.0 degree (i.e., inclination = 89 degrees). We must associate the inner orbit collision (and the fragments created) with brief fade events, and we must associate the big outer orbit collision (and those fragments) with the slowly developing long-term fade of out-of-transit (OOT) brightness. Notice that for an inclination of 89 degrees the distance between our line-of-sight and the two orbits are 0.0035 and 0.035 a.u., respectively (orbit radius x sin (90 - i)). These are the distances that an expanding dust cloud will have to achieve in order to block starlight from our view. If the dust cloud expansion velocities were the same (just a starting guess) then the outer cloud would take 10 times longer after a collision to begin blocking starlight. During that time it will have been sheared to all longitudes of the orbit. The dust cloud would soon take on the shape of a torus, with a constant average orbit radius but with an expanding cross-section. In other words, blockage of starlight by the outer cloud will not vary on timescales much shorter than the 900-day orbit. All of those torus blockages of starlight could have been set in motion by one big collision (with a planet at 2.0 a.u.).

The inner cloud, however, will be more changeable for two reasons: 1) the interval between a collision fragment's production of dust and the time it reaches our line-of-sight to begin blocking starlight is much shorter (1/10 of that for the outer orbit, assuming similar isotropic velocities), and 2) the dust temperatures for the inner dust cloud can be hot enough that small particles will sublimate and become a gas that doesn't block starlight (except at very specific and narrow wavelengths). This last point means that the inner dust clouds can come into existence and then disappear (due to sublimation) on short timescales, possibly comparable to their orbit period of 30 days.

Notice that I referred to "fragments" from an inner orbit collision as sources for the dust clouds. I am suggesting that the brief but deep fade events are caused by fragments from possibly one collision with a planet in an inner orbit (0.2 a.u.). These fragments will have periods similar to the planet from which they were broken off, and they will be the source for dust clouds producing fade events.

The other collision, with a planet in the outer orbit (2 a.u.), would not have to occur at the same time as the inner orbit collision. The outer orbit collision could have occurred centuries ago. In fact, it probably had to have occurred at least a century ago in order for the dust from the collision to become stretched-out into torus shape, and for the torus cross-section to have expanded enough to enter our line-of-sight to the star and begin a long-term fade.

Consider again the expanding torus of dust in an outer orbit. It will begin to produce a fading that starts long after the original collision (given by orbit radius x sin (90 - i) / dust ejection speed). As the cross-section of the torus continues to expand it will become less dense along our line-of-sight, and thus begin to block less starlight. The amount of fade by this long-term component will be determined by the line-of-sight column content of dust (and involving a size distribution spectrum).

The above physical model has features that are compatible with observations of both the brief fades, lasting a few days and capable of being deep, as well as a decoupled long-term fade that is gradual in its rise for blocking starlight and gradual in its recovery to negligible blockage, with characteristic timescales of years. [For an extensive discussion of fragment-produced dust clouds that produce fade events orbiting a white dwarf, WD1145, read https://arxiv.org/abs/1608.00026 ]





Comparing Kepler with Recent Light Curves

Here's the last 140 days of the Kepler light curve. It shows the deepest dip observed, 21%, Dip #8 at Kepler Day 1520 (abbreviated as d1520). The second deepest dip, Dip #10 with depth =8% is at d1568. The dip at d1540 (Dip #9) has generated interest due to its symmetric pattern of a 3% dip flanked by 1.1% dips on either side, separated from the main dip by ~ 3 days. Many people have been attracted to the idea that Dip#9 is produced by a large planet with rings, and it's the rings that produce the two flanking dips.


Figure 3.1. Kepler light curve for the last 140 days of TS observations.


Figure 3.2. Same data on expanded date and normaled flux scales.


Figure 3.3. Comparison of Kepler and Gary light curves using similar date and normalized flux scales. [The observations panel is not up-to-date; I'm not a believer in this association so my motivation to update is low.]

The similarity of structure for the Kepler Dip #9 and the recently-measured Jun 16 dip is apparent. Even an earlier dip (before Kepler Dip#8) has a correspondence with the May 19 dip. Only the Kepler d1520 feature (21% depth) is missing in recent data (maybe it's a dust cloud in a different orbit than the Kepler d1512 and d1540 features). I'm not bothered by the difference in depth of the Kepler d1540 dip (3% vs. 2%) because dust clouds should disperse over time, and produce smaller dips. This is an interesting similarity of pattern (but I'm not a "believer"!


Long Term Trends

My original purpose for starting KIC846 observations 1.7 years ago was to measure long term trends; catching fade events wasn't my goal. I think I've measured a long term fading trend, and it seems to be "accelerating."

If dip measurements are excluded from consideration there are long-term trends in TS "out-of-transit" brightness. Here's a plot of my V-mags during the past year. Note the slight curvature for the model fit.


Figure 4.1. V-mag's for TS during the past 300 days. Filled circle symbols are judged to be "out-of-transit," or made when dips were not underway. Open circle symbols are the dip data. The late 2016 data exhibits a 29-day periodicity.


Figure 4.3.  A 1.7 year's worth of V filter and clear filter measurements has been fitted with a model. Only OOT data are shown. The clear filter magnitudes were adjusted to match near-simultaneous V-mag measurements. A Gaussian model was fitted to both sets of measurements. This speculative model "predicts" a return to "normal brightness" after several years (after 2021).  

The Gaussian model is meant to represent a dispersal of dust from a giant collision that produced dust that is slowly being spread out along the orbit where the collision occurred. The dust would initially produce brief and possibly deep dips, but as it disperses the dust will be present at all parts of its orbit and cause a fade that persists for as long as the dust is present in the orbit plane where it can intercept our line-of-site to TS. This is just a speculation; I have no idea if this is correct.

A Prediction

I predict that in a matter of months the brightness of KIC846 will NOT plummet to zero, but the level of interest in KIC846 by the general public will! This will happen when the professional astronomers present evidence that the fading events are produced by dust clouds, not alien mega-structures. The hit rate for my KIC846 web pages, which is currently about once a minute, will return to the once a month level, just as they were for the 1.5 years before May 19 of this year.

I have experience with the fickleness of such things based on my 2013 observations of Comet ISON - that so-called "Comet of the Century." NASA bought into that terminology, possibly because it garnered public interest in a project that NASA had funded. I was the first to dampen enthusiasm when I made the first "recovery" observation after a 2.5-month hiatus of imaging (due to the sun being close to the line of sight) as the comet was making its approach, and I stated that the comet was about a magnitude fainter than models were predicting, and it might disappoint during the rest of it's approach to perihelion. NASA continued to hype the comet, and I attracted a fan club of cynics who trusted my assessments and almost daily updates more than those by the professionals. Some unscrupulous web hucksters took pictures from my web site for posting on theirs with hyper-hysteric interpretations; one posted a picture I had enhanced to show a forward jet and on his web site he claimed to show a UFO flying in formation ahead of the comet. As perihelion grew closer, and the comet's brightness fluctuated at below expectations, and as my web page hits grew to ever higher levels, and as experts were quoting my kill-joy assessments, Discovery Channel scheduled an interview with me at my observatory to coincide with perihelion passage. But as the sun's heat sublimated the comet to nothing during closest approach; public interest immediately plummeted to zero and the the Discovery Channel interview was canceled. My web page hits went to nothing, and my life returned to normal. But I was wiser, for I finally understood the fickleness of public interest in things scientific.

I predict that by the end September, when the evidence for dust clouds and prosaic decadal variations are accepted, KIC846 will join the "Hype of the Century Club" for being "the alien mega-structure that wasn't."

References

    Simon, Joshua D., Benjamen J. Shappee and 6 others, "Where is the Flux Going? The Long-Term Photometric Variability of Boyajian's Star," arXiv:1708.07822 
    Meng, Huan Y. A., G. Rieke and 12 others (including Boyajian), "Extinction and the Dimming of KIC 8462852," arXiv: 1708.07556  
    Sucerquita, M., Alvarado-Montes, J.A. and two others, "Anomalous Lightcurves of Young Tilted Exorings," arXiv: 1708.04600   Also, see New Scientist link and Universe Today link.
    Rappaport, S., A. Vanderburg and 9 others, "Likely Transiting Exocomets Detected by Kepler," arXiv: 1708.06069 
    Boyajian et al, 2015, MNRAS, "Planet Hunters X. KIC 8462852 - Where's the flux?" link
    Ballesteros, F. J., P. Arnalte-Mur, A. Fernandez-Soto and V. J. Martinez, 2017, "KIC8462852: Will the Trojans Return in 2011?", arXiv
   Washington Post article, 2015.10.15: link
    AAVSO Campaign Notice requesting KIC646 observations
    AAVSO LC Generator https://www.aavso.org/data/lcg (enter KIC 8462852)
    Web page tutorial: Tips for amateurs observating faint asteroids (useful for any photometry observing)
    Book: Exoplanet Observing for Amateurs, Gary (2014): link (useful for any photometry observing)
    My web pages master list, resume


    B L G a r y at u m i c h dot e d u    Hereford Arizona Observatory    resume 
 
This site opened:  2017.06.18.  Nothing on this web page is copyrighted.