Timing of next big Dip

When will the next Dip appear at Boyajians Star?

In the last posts (d792, d1519), I described a model of the different large dips. This model is based on a beam that lifts matter from the star surface into an orbit. Due to the geometric situation, it should be possible to calculate the rotation speed of the beam. Actual the rotation speed is part of the parameter set in the shape calculation of the beam.
In this post, I will suggest a possible period.

The parameter rotation speed

The timing of the beam calculation is based on the equation

α(t) = ((BJD-2454833) – t₀ ) ω

BJD-2454833 is the time for all the Kepler data [1], based on the BJD time,
t0 describes the dip at day 792 and has the exact value 792.7216d
ω is the rotation speed and has to be fitted to the data.
If we know the exact value, we caught the rotation period of the beam and can calculate the reappearance of more dips.

In the first paper, I suggest ω has the value of 1.00E-02[1/d]. This correlates to a period of circulation of 628 days (2pi*100). The loosely guess of the 100 was done with no special precision.
In a review of the situation, I changed the value from 100 to 115,67 (Distance from d792 to 1519) and checked the effect to the shape of the calculated dips, with a special concentration on the lower part of the absorption. As shown in figure 1.

Figure 1: Dip 792 optimized right part

This fits very nicely in the in the area where the error was considered, shown as red error bars. A calculation when I change the period by one-day results already in a higher error in this region.

The other part of the plot is not that perfect, but other factors are not optimally implemented in the model, like the beam cross section.
To remember, in the right part under investigation, only the beam generates a shadow, the shape of the beam is not as influential as in the area of the deep dip.

Dips in the context

If we use the flux calculation for the whole Kepler period, we can see the repetition of the calculated flux as shown in figure 2:

Figure 2: Flux over the whole period (Right scale "Calculated Flux" lowered for good readability)

We see of course the tick at d792, which is the base of calculation. But we see also a dip at day 429, and day 1156, both are in some way an artifact, because the equation works with the absolute sin and contains in this sense a beam that would be on the other side of the star.
But there is an interesting detail, very near to this dips, we see in each case a small dip in the measure of Kepler. Let's have a look at the detail of dip d429 in figure 3:

Figure 3: Timing of d426 and the timing of the dip artifact.

It might be, that the technology behind star lifting generates some small beam at the opposite side of the star, this might be due to a magnetic field. But this can also be a pure coincidence. Due to the fact, that at day 1150 another similar dip is visible, should focus some resources to understand this coincidence.

The second appearance

At day 1519, we see the second appearance of our dip d792. The shape has changed but an interesting detail is still there, look into figure 4:



Figure 4: The dips around day 1519

In this plot, the shape of the d792 reappears due to the period of 726.78 days. Other elements add to that dip as discussed in the post 1519, but there is one amazing element, at day 1518, the flux recovers just to the level of the remaining effect of d792, but not more. This might be interpreted in a way, that this is, in fact, the separate structure of dip 792 appearing together with other elements of beams which reduce the flux at other times in the interval.

When will the next Dip appear

The most interesting question is, when will the next Dip appear?
I make the prediction as following in a table:


Time of dips in UTC time, Kepler starts at t = 0 (BJD-2454833): 2009-Jan-1 11:29:59, I used the converter from Ohio State University

  dip      BJD              UTC
   -   2454833      2009-Jan-1  11:29:59 (Start of Kepler time)
  792  2455625.722  2011-Mar-5   4:49:40
 1519  2456352.500  2013-Mar-1   0:35:59
 2246  2457079.278  2015-Feb-25 18: 4:19
 2973  2457806.057  2017-Feb-21 13:58: 4
 3700  2458532.835  2019-Feb-18  7:26:24
 4427  2459259.614  2021-Feb-14  3:20: 9 

(I am not completely sure about the time conversion, if someone is here an expert, he may check the Kepler start time) I have now revisited the timing conversion, using Eastman, Jason tool [2] and U.S. Naval Observatory Astronomical Applications Department the result is slightly different but very well within the error margin. Tuesday, A.D. 2017 Feb 21, 13:18:25.1 (JD 2457806.054457)

On February 21st should see a dip!
 I am very curious to see if the result matches the prediction. 

I am very curious to see the result.

Reference:
[1] http://archive.stsci.edu/kepler/
[2] Eastman, Jason; Siverd, Robert; Gaudi, B. Scott, 

Achieving Better Than 1 Minute Accuracy in the Heliocentric and Barycentric Julian Dates

DOI 10.1086/655938

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.

Dip Day 792 in Detail

Calculation of Dip 792d with Star lifting

This post is the second part of the calculations to understand the shape of the dip at day 792 of Boyajian's star (KIC 8462852). For part one visit post: "Do we see Star lifting".

The Beam is Bent

The Dip day 792 is not symmetrical, therefore we have to introduce a beam, that is not perfect on a straight line from the center of the star. it is not difficult, to find reasons why the beam of the matter is bent in a direction. A simple reason is in Kepler dynamics during departure from the surface. 

As long as we don't understand the process in detail, I will again start with a simple model and try to fit the model with the data.

Figure 1 describes the new model of the beam. The beam starts at the surface of the star, and with height H it starts to have an optical density due to reduced temperature and therefore is no longer ionized.  

Fig 1: Properties of the physical model.

The beam is now modeled with small fractions of the length ds which are distorted against the direction angle depending on the distance l. At the end of the beam, the "smoke" comes to a rest and is distributed in this orbit by an exponential law, the density is maximal at the beam end and decays with distance in the bending direction.
For the simulation, the beam contains 105 elements. Each represents a beam length of 5 star radii, The first 5 star radii (first element in the simulation) are transparent due to the high temperature near the star surface.
Each simulated beam element has a transparency of 0,9984722. Each element is a little bit bent by
ap = -0,3971 day.
If an element is within the line of sight between Kepler and KIC 8462852, the resulting transparency is calculated by the multiplication of the transparency of every element.
At the end of the beam, the material enters an orbit, for some reason, and is accumulated. The optical density of the accumulated material decays by an exponential function d = d0 exp(a*w) thereby d0 = 0,001588, w = 3,30025 [1/day]. (The unit hour is used and could be converted in an angle if the distance and rotation period would be known). The value of the simulation was always averaged over 2.5h to adopt a little bit the shape of the unknown optical elements.
The result of this simple physical simulation is shown in fig 2:

Fig 2: Measured and simulated flux using the model

The model, containing only simple and plausible parameters fits within the limits of the Kepler data precision. The value of the free parameters, namely bending, optical density and the exponential function, were optimized manually.

I did not expect a result with such a low error.

There are some aspects not included in the simulation: an exact cross-section of the beam, surface flux of the star (it was assumed to be constant).

Other posts related to Tabbys Star

Dips part one

Dips part two
A simple model of Dip 792

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