A deep dive into Dip 792 part II

Calculating the residual Signal

In my last post, I started to analyze Dip d792 in detail. But my work was suspended due to a very exciting new dip that has been observed in May 2017.

The first part is here A deep dive into Dip 792 part I.

Mathematical description

The remaining signal might be the result of a fog cloud, that is in an orbit around the star.

The intensity of this cloud could be calculated if we subtract from the measured signal from Kepler the basic dip. If we choose the timing of the symmetric axis a little bit shifted, to take into account the cloud in the area of the peak of the dip, we find two very smooth exponential decays, as shown in the next graph:

The remaining signal shows two exponential growth factors.

(missing data points are marked with black dashed circle lines)

Let's look into the details of the plot:
All data are relative to the central peak in a convenient time scale t = t_k-792.740 d (Zero is at the peak of dip 792)

• The raw dip data from Kepler are marked as black vertical x marks.
• The assumed basic dip is presented as a black line and is the mirror of the right half of the dip, not shown in this plot, both data rows use the left linear scaling.
• The remaining density is shown in a logarithmic scaling as shown at the right axis. The first part with very noisy data is presented as grey dots and not further processed. 
• The middle part is presented by violent squares and can be approximated by an exponential function with a starting value of 0.022 and a growth factor of tau1 = 0.5298 per day.

This part changes obviously the direction at t = -0.633 d to a steeper growth with a growth factor of tau2 = 1.6655 per day, presented as green x marks.

Both approximations have a very high coefficient of determination, R² = 0.9809 for the first part, which has more noise and R² = 0.9953 for the second steep part.

Here comes a high zoomed plot of the second steep part:

The measured signal x marks and the calculated function as circles match in the time range -0.6d to 0.2 day within  0.1% as shown as vertical x marks in the center.

The second plot is focused on the steep function (green in the first plot) in the second part, the deviation is below 0.1% over 0.5 days.

It should be mentioned, that the ratio between the growth factors is 3.143... for whatever reason. (I think it is by pure chance because any intelligent species would use 2 * PI = 6.28... to signal their math level).

But beside this strange coincidence, the exponential decay could have some physical reason. It might be, those dust particles stick together, brightening the "dust" in the orbit.

The two different growth factors may result from two different processes with different efficiency.

Has anyone seen something like this in a comet tail?

Thank you for your attention and give me feedback if you have any comments on this.

Dip 792 at Boyajian Star KIC 8462852 revisited

A deep dive into Dip 792 (part I)

The dip 792 of the most mysterious star in the galaxy has a very regular shape and is different from all other signals seen in the Kepler mission. (If new, find an introduction at Dips in Tabby's Star)

The famous Dip of Day 792 at Boyajian Star (Data-Source: NASA Kepler Mission)

What makes the Dip so unique?

The first thing is, this dip is very deep, about 15% of the light of the star is invisible to the telescope during the peak of the dip. This is a very high number, when we compare that to a transit of a Jupiter like planet, which would absorb about 1% of the starlight and when we remember, that Jupiter is near to the maximum size of a planet.

The second thing is the shape, it is definitely not the shape of a planet, as has been shown by Ksanfomality [2], The asymmetric structure does not match with a high eccentric planet due to timing and absorption. The same is true for comets, the density of a comet and the homogeneity of the dip is not something we expect to see by well-known comets analyzed in our solar system.

The third mystery is the obvious smoothness of the dip, this is only possible if a relatively simple shape or process is the cause of the dip. we should be very happy to have this dip 792 near the other much more complex and hard to understand dips later in the Kepler mission survey. If we understand D792, we might be able to take the hints to comprehend the other dips.

The fourth mystery is the total time of the dip, that is about 8 days. No massive object, even another star, would occlude this star for so long.

Analysing the shape

In this post, I do not try to give a possible explanation of the shape as done before [3][4], but I look into the characteristics of the shape.

End and Start of the Dip

The first tricky thing is, to determine the length of the event. The end of the event seems to be easy to determine using a graphical extrapolation. If we look in a high-resolution plot of the data we find, that the dip ends at day 794.85. The uncertainty is within an hour or a few measurement points (0.5h).

Dip 792 ends day 794,85 within a well-defined uncertainty of a few measurement points.

The begin of the dip is much harder to define, due to the very smooth beginning of the dropping light intensity. It seems so, that the density of the absorbing object is fading away with no sharp border as seen in the high-resolution image, be aware of the slightly wider timescale.

Dip 792 starts with a very smooth slope of the signal at day 787.

We might get a better understanding of this shape when we assume an exponential decay as shown later in this post.

Missing Datapoints

A look into the measured brightness at the recovering brightness at Day 792 shows a lot of missing data points. This is more an instrumental issue, but it should be mentioned. For further calculations, the missing data points were substituted by calculated values, using a linear interpolation of the value of the points before and after the gap.

Missing data points were calculated by linear interpolation using the two data points at the border of the gap.

The steepness of the curve at the most dramatic part is 0.5%, that compares to a Jupiter sized object, that is leaving the solar disc within one hour. 

Symmetric Dip?

The shape of the dip is not symmetric, but what, if we assume, there are two parts, one symmetric event (basic dip) and another part (fading part) which is adding to the symmetric part and resulting in the visible dip? 

This calculation can be done if we assume, that the first part of the basic dip has the same shape as the second part. The second part is well known and the only thing is, to generate a mirror picture with a meaningful axis. To generate an easy to read mirror image, the missing data points during the measurement of Kepler were substituted by a linear interpolation of the available border points of the gap, in most cases, only one point was missing, resulting in a minor error.

The result is visible in the next plot, where both plots are visible. Blue is the original time series, red is the same graph, mirrored at the axis at the time point 792.73d. 

The symmetric assumption of the dip 792.

We can now synthesize the basic dip and the fading part. First look at the basic dip:

The basic dip is an artificially constructed dip using symmetric assumptions.

The basic dip has now a defined start and end, using the point of symmetry, 792.73d, and the value found for the end of the dip at 794.85d. We define this information:

Start of basic dip: t1 = 790.61d

Maximum: ts = 792.73d

End of basic dip: t2 = 794.85d

Duration: T =  4,24d

Further analysis of this shape might be interesting, but is not part of this post, maybe I will discuss this in another post.

The remaining Signal

Very interesting is now the question, what happens to the remaining signal. The remaining signal, calculated by subtracting the basic dip from the measured value, is shown in the next plot:

The remaining fading part, if we subtract the artificial basic dip from the measured data. 

The fading part seems to have an exponential character. This should be analyzed further.

The best way to see the exponential character is, to plot the data in a logarithmic scaling in the absorption axis. 

To optimize the result, the data before day 791 are reduced in noise, by not subtracting the symmetric signal but by using a constant baseline with the value one. Otherwise, meaningless noise and fluctuations from the baseline beyond the basic dip would appear.

Log-plot of the absorption in the remaining signal after subtraction of basic dip.
Different colors mark different slopes.

The log-plot shows at least four different areas. At the left part, there is a lot of noise and a relatively steep slope. The blue squares mark an area with a constant slope, the exponential factor is 0.46 1/day. Around day 92 the slope increases significantly to a higher value of 1.32 1/day. The last points at day 792.55 and following, don't show any pattern and may result from a remaining error concerning the unknown exact shape of the basic dip.

Here comes the second part.

2017 May 19: We have a new dip, I publish this, and wait for new results.

Live: https://youtu.be/eYpIGZS8nJc

Dip Day 1519 in Detail

Solving the puzzle of Dip 1519

This post continues the analysis of different dips seen by the Kepler telescope at Boyajian's star (KIC 8462852). To understand the discussion, I recommend reading the analysis of Dip 792, because I use the same basic model.

Again a Starlift model

The very useful stairlift model of the last post is reused to understand the very complex signature of the dips around day 1519, as presented in fig 1.

Fig 1: Komplex deep Dip 1519

The dip includes a very deep double dip, with 22% absorption, and an asymmetric basic structure, similar to dip 792.

My idea was, to use the simulated shape of a starlift with "smoke" as described in detail in Dip 792, to understand the shape 1519. Therefore the shape was positioned three times in the time frame with a different position in time and different absolute absorption.

The result is shown in fig 2.

Fig 2: A first attempt to reproduce dip 1519

The simulated black line does not reproduce the blue measured values, but some very significant elements are well done. First of all, the double asymmetric main dip fits just perfect. And this was done by simply adding the model of Dip 792 with the same intensity, the factor is in both cases exact 1.0 (one)!

The first deep dip is deeper than the second, the reason is, that the "smoke" of the second dip deepens the first. The numerical distance in time was set to 5h. The length of the starlift beam is 1.50 higher as in dip 792. This also means that the orbit of the smoke in this simulation is by a factor 1.50 further away as in dip 792. 

To complete the picture, a third starlift beam was introduced 12.5h before the main dip.  

The factor for this dip is 0.36789. Ever seen this number? It is 1/e, but it seems so, that value is by accident similar.

e is the very well known Euler number 2.7182.... the number of natural growth and the mathematical basis for the success of any civilization. (More about at wikipedia)

Some Problems

But the simple model does not reproduce the measured line in all parts. In area A, B and C, the signal is brighter than it should be. There are two possible explanations:

1. The starlift beam had some interruptions and so the smoke has some breaks.
2. The material of the smoke was used for construction and is no longer at this place in the orbit.

If we like solution 2, then it is not to hard to understand bumper D in fig 2. It might be some material in the orbit, it could even be a mirror to power the starlift itself. But this is pure speculation. 

A better solution is given as a hard puzzle to the reader.

Thank you for reading and please give me feedback. 

Hopefully, a paper, concerning this research is soon completed.

Do we see Star lifting

The Try of an Explanation of the Dip at day 792

In the last time, there has been the suggestion, that the very strange dip around day 792 might be the signature of star lifting. A reasonable explanation of star lifting can be found at Wikipedia.

Fig 1: The dip at day 792 has a very interesting homogenous shape.

The basic Idea is, that a super civilization is able to harvest matter from the local star by magnetic or other means. This is quite difficult, due to the high temperature at the surface of the star. Therefore a beam of matter, similar to a natural solar solar flare has to be produced. I don't go into the technical details how or if this is possible, but I try to simulate the visible lightcurve of such an activity. 

Fig 2: Natural solar flare at our Sun.

The most simple model

(A more complex model is described here "Dip 792 in Detail")
To generate the situation, we start with a very simple model. A long beam from the star points radially away from the center of the star. by accident, we are in the line of sight and see the beam crossing the star.

Fig 3: Simple model, a beam of matter, pointing away from the star.

To describe this model, we assume, that the beam is rotating around the star and a fraction of the beam absorbs the starlight on our line of sight.

The amount of dimming is then a function depending on geometric factors and the rotation angle as shown in fig 4 viewing the situation from the rotation axis of the system. 

Fig 4: Geometric situation, Kepler looks from the right side to the star with center C.

The star has a radius of r and the center is marked by C, a beam with an optical density d, at the surface of the star starts at point A and ends at point D. The dimming is proportional to the length, of the distance |AB|, because only this part of the beam covers our line of sight. We can calculate the distance CB, depending on the angle a, it is

|CB| = r/sine(a)

len(AB) = r/sine(a) - r  (1)

The angle a is depending on the time t, and the angular velocity w, by which the beam rotates around the star like a hand at a clock. 

It is convenient to set the time t to zero when a = 0. To suppress the infinite length of AB at a = 0, we have to take into account, that the beam is not infinite, but has the length AD. The equation (1) holds therefore only as long as the beam does not cross the sightline BE. This happens at the angle 

ac = arcsine( r / |CD| )

ac = arcsine( r / ( |AD| + r ))  (2)

The measured flux f(t) is then calculated, assuming f0 is the brightness of the star, by

f(t) = f0 - d ( r / sine( w t ) - r)   in the case {||wt|| > ac}

f(t) = f0 - d |AD|   in the case  {||wt|| < ac}   (3)

Let's have a look at our dip at day 792:

Fig 5: Very simple model of star lifting. (Sorry, for some reason this image is flipped in time)

Although we have used the most simple model, the left part of the graph is astonishingly similar to the measured dip. The parameters used for equation (3) are d = 0.16, f0 = 1, r = 1, w = 0,14 [1/30min].

An inhomogeneous absorbing beam

A further approximation to the real situation can start with the optical density of the beam. At the surface of the star, the beam has the same temperature as the solar surface and will not absorb any light. By leaving the surface the beam cools down and the atoms might absorb light by ionization. Again, I try a very simple model, the temperature is then depending from the visible surface of the sun, depending on the height. 

My geometric model is plotted in fig 6.

Fig 6: Temperature of the beam as a function of height |AB|

At the point B the beam has a height above the star of |AB|=h. The visible angle a is therefore given by the geometric relationships in the rectangular triangle BCD.

|AB| + r = |BC| 

and

sin(a)=r/|BC|

sin(a)=r/(|AB| + r)

a = arcsine(r/(h + r))   (4)

The visible cone P is, therefore, relative to the 2 pi surface situation

P = 2 pi sin(a) /2 pi

P = (r/(h+r))   (5)

A plot of this function is shown in fig 7.

Fig 7: Showing the surface of the star, a beam sees at a distance from the star

To keep things simple, we define a height, where the reionization happens. The function in fig 7 drops fast and the black body radiation is depending on the fourth power, so I guess with a height of two-star radii, the beam is reionized. 

To include this in our model, we replot a modified figure 4 in fig 8.

Fig 8: Modified beam with optical density starting at H

In the new model, the beam starts to have an optical density, starting at the point H and going up to D. Using this, we can replot the simple calculation from fig 5 again, only presenting the falling edge for better visibility:

Fig 9: The model matches the measured flux better in the first part.

Although the model is still very simple, the match between measured flux and model is now also in the first part good. Be aware, that the ionization doesn't happen instantly and has to be modeled by a more sophisticated model the basic effect of a certain threshold seems to exist. 

The steep side of the dip is not perfect, this may be due to the inhomogeneous radiation density of the star, another point that a better model should include.

In the next post, I will try to model the very steep rising edge by bending the beam. 

If anyone wants to support me with efficient computer models, he is invited please drop an email heindl(a)gmail.com

Other posts related to Tabbys Star

Dips part one

Dips part two
A more complex model of Dip 792

Meditation over the WTF Star Dips, Part II

KIC 8462852 and the really deep dips

This is the second part of my meditation over the strange dips in the light curve of Tabby's Star. You find the first part here: "Some aspects of KIC 8462852".

Dip 9 

The following dip 9 is not very spectacular, the signal is near the noise limit. 

Fig 1: Dip 9 at d848

Be aware, that most of the signal in this plot is due to the rotation of the star, period 0.88 days, and the fluctuation of the brightness is, as far as we understand other stars, a result of sun spots.

Beside this very natural signal change we see a dip, that starts at day 846 goes down to a minimum near day 848 and then the brightness recovers again. The exact shape is not known, due to the noise and influence of the sun spots. The shape in depth and time is not unlike a small planet, similar a transit of the earth in front of our sun. 

Dip 10 and 11

The dip numbers, to be exact the time frame, are automatically generated by the computer. The size of any signals in this period is not useful for any further discussion. May be we find a periodicity then it could be a hint for any object like a planet.

Dip 12

In the case of dip 12a we have no information for the dimming part, due to some measurement errors. 

The recovery of this relative small dip, 0,11% depth, shows a unusual behavior for a planet transit. But it has a very similar shape as dip 8, going back to normal brightness with some type of exponential looking shape.

Fig 2: Dip 12a at d1126

The strange thing with this dip is, why do we see the measurement error at the beginning and then an exponential recovery? We can only understand the quality of the shape, if we understand exactly the reason, why we see a measurement error. If anyone reads this blog with more background on the detector system of Kepler and this glitch, he is welcomed to give me a hint.

The case of dip 12b looks again structured.

Fig 3: Dip 12b at d1143

The dip 12b has a deepth of 0,12% and has a ramp before a very steep dip follows. this is a little bit similar to dip 1, although there is much more signal available. Then follows a floor as already seen in dip 2, and then a similar ramp to recover from the dip. A planet with a accreditation disc might show a similar shape, the problem is the timing. The central dip lasts more than two days, this is hardly possible for a planet orbiting a large star.

Fig 4: Dip 12b with manually interpretation of the shape

Short, within the ramp of dip 12b follows dip 12c, a small dip with a typical shape of a planet transit,

Fig 5: Dip 12c at d1151 lasts about one day

The duration of one day is about 2 times shorter as the central part of dip 12b.

Dip 13

 Dip 13 may be a member of another class of dips, looking very symmetrical. But it could also be interpreted as a case of 3 to 4 consecutive dips, which are by chance similar in size.

Fig 6: Dip 13 at d1205 a very symmetric shape

We could compare this dip with dip 4a, also a set of dips, that starts with a small dip, then a center dip and then another small dip. There has been some discussion about timing and depth within this dip.

Depending on the baseline, there could be a rational 1:2 between the minor dips and the major dip in the center.

Fig 7: Dip at d1205 with a projection of the mirror of the image.

I will look in the interesting symmetric shape of the dip. Therefore I include the mirror image into figure 7. We could see at least four elements, A, B, C, D, ups and downs in the flux, which apperare with perfect timing relative to the central symmetric line. It is hard to believe, that this happens by chance. Some physical reason could be a ring system around a planet. But the shape of the central dip does not support this idea. Very strange is, that dip 16d at day 1536 has a similar shape, but a different size, I will discuss this later.

A Planet?

If we look into some details of the complex dip structure, it should be mentioned, that at d1208,2 a small rectangular dip shows up, similar do other dips like 12c at d1143. A time difference or 137 days.  Adding this, the next expected dip should be at d1417 and the at d1554. At d1417, Kepler has no signal due to technical problems, but at 1554 there seems to be the same dip (depth delta 21 [e-/sec]), may be the same object! There is also a small dip at 1007, nothing at 869, no data at 732, and 321, but a small signal at 185. Planet hunters should look into the details.

Period 14 and 15

Due the time interval from day 1274 till day 1471 no very significant event appears. It should be mentioned, that at d1433 a drop in the flux with the typical duration, often seen before, of 8 days but with a small amplitude 38 (s-/sec) is visible.

Fig 8: Dip at d1433, amplitude in the range of typical fluctuations due to sunspots(?).

Period 16 and 17

Period 16 and 17 contain the most dramatic fluctuations ever seen in a star of this type. The flux is up to 22% dimmed. Very hard to understand by well known astronomic events. The shapes seem to be part of one larger, symmetric event, as pointed by Gary D. Sacco in the reddit thread "95 Day Abnormal Equilibrium of Periodicity and Flux Variation" [1]

Fig 9: a 95 day period with more or less symetric deep dips, source gdsacco [1]

As Gary D. Sacco points out, the distance between the different dips seem to be arranged near a central dip at day 1539. The shape of dip d1539 (depth 670 [e-/sec]) is visual similar to dip 13 at d1205 (deepth 111 [e-/sec]), although 6,03 times deeper.  

The symmetry is by far not perfect and the optical center of the dips is not the center of symmetry. Very strange is the aspect, that the sequence before the first and second large dips are a little bit similar on the time axis. If we look into the structure of d1519 in fig 10, we see a complex structure.

Fig 10: The left large dip at day 1519 seems to be a double dip 

To get the details, I added some lines see fig 11. 

Fig 11: The elements of dip group around d1519.

After two small dips A,B, the flux recovers and starts to drop strong along a line C. Another object D comes into the scene and accelerates the drop. The flux recovers a little before dip E and F come in. For some reasons, after dip F recovers, a linear flux reduction, given by line G appears this might be part of another element that is wider than the object that resulted in E and F. Object H might produce the next dip and another, different thing, shown by line I leads to the final recovered intensity.

Together at least nine different elements of whatever nature result in the strange flux signature.

Now look into the second big dip:

Fig 11: The right dip at day 1568 has a single full dip.

I also try to introduce some helpful lines shown in fig 12:

Fig 12: The elements of the dip group around d1568.

It starts with a slight decay of the flux during period A, and a first dip B five days before the full dip F shows up. But before that after s light lower plateau, again a slight decay C and then the first large dip D, and similar to d1590 dip C, and then a stronger dip E, similar to D in d1590. After a slight recovery the main dip F appears. During return to normal, a small dip G appears, might be in some way similar to dip H in d1590.

The choreography is similar, the values are not similar and they are not simple mirror signals around the symmetric center. This makes the understanding of the reason of the signal much more difficult. For example a disk like Saturns rings could not explain the flux.

It should also be mentioned, that the choreography is by the structure similar to dip 7 d694, although this dip is are much smaller in depth.

Conclusions so far

• The different dips seem not to be from the same family of natural events, of whatever type they are. 
• Interestingly, most of the dips have an internal structure. Only very weak dips don't show a visible structure, but this is a effect of noise, we are not able to see them.
• Most events have a similar choreographic structure, they start with small events and the biggest dip is at the end of the event, a little like in a firework (:-).
• There are at least two events d1205 and d1540 (and d359 a little) that show an astonishing symmetric structure, that is hard to be explained by accident, like comets coming with the right timing. Although there might exist natural explanations like ring systems around a planet, which could lead to such a flux variation.
• The strangest of all dips is d792, it seems not to be the product of a multi event. My best guess is, that something pointing away from the star, like a column of smoke, is the cause. This is implied by the tangent function and the exponential function, that fits very well (see first part). If the "smoke" is a little bend to one side, even the asymmetric structure can be plausible explained. 
• The reason of the smoke column could be a internal event of the star, similar to a solar flare, but millions of miles high and cooling down. The artificial star lifting should be included into the discussion.
• All events seem to happen within a time frame of less than 10 days and last at least five days. This tight time frame is another strange independent element of the KIC 8462852 story.

back to part one of the meditation over WTF Star dips

Next part is the dip of day 792 a sign of star lifting, the post contains some mathematical analysis.

References

[1] Gary D. Sacco, https://www.reddit.com/r/KIC8462852/comments/56kdfw/95_day_abnormal_equilibrium_of_periodicity_and/