Fourier Series and Cepheid Variables

  • #1

Summary:

If given a set of data points for the magnitude of a cepheid variable at a certain time, how can we use fourier series to find the period of the cepheid variable from the equation of the light curve? Where can I find data points for M31_V1?
If given a set of data points for the magnitude of a cepheid variable at a certain time (JD), how can we use fourier series to find the period of the cepheid variable?

I'm trying to do a math investigation (IB math investigation) on finding the period of the cepheid variable M31_V1 from data points for the magnitude of the cepheid variable at a certain time (JD). However, to find the equation of the light curve using fourier series, the period must be given already right? How can I find the period then?

Also, I'm trying to plot the data points from the AAVSO (https://www.aavso.org/data-download/aavsodata_5f913c1d4f3f0.txt), but it seems like the points are not forming a curve. Can I find data points for the magnitude and JD of M31_V1 anywhere else?

Thank you!
 

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  • #2
mathman
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To get the period you would data at a series of times. I don't understand what you have in mind.
 
  • #3
To get the period you would data at a series of times. I don't understand what you have in mind.
From this paper https://arxiv.org/pdf/1111.0262.pdf
under 'analysis', the period is said to be obtained through fourier series. I didn't quite understand how that was possible
 
  • #4
To get the period you would data at a series of times. I don't understand what you have in mind.
Quite honestly though, I'm feeling very lost about what my focus should be for my mathematical investigation.
 
  • #5
mathman
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The data points are magnitudes at many times. The fluctuations give the period.
 
  • #6
phyzguy
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Perhaps you misunderstand the difference between a Fourier series and a Fourier transform. If you have a series of data points with some quantity vs time and you take the Fourier transform, the Fourier transform will have peaks that tell you the periodicity of the variation. If it has a single period, it will have a single peak. The data you linked for M31_V1 should be sufficient to do that analysis. The observations typically can't be taken as often as you want because of clouds, daylight, the moon, etc. But a Fourier transform will still be able to extract the period, even though you may not be able to see it in a simple graph. That is one reason why it is such a powerful technique.
 
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  • #7
Perhaps you misunderstand the difference between a Fourier series and a Fourier transform. If you have a series of data points with some quantity vs time and you take the Fourier transform, the Fourier transform will have peaks that tell you the periodicity of the variation. If it has a single period, it will have a single peak. The data you linked for M31_V1 should be sufficient to do that analysis. The observations typically can't be taken as often as you want because of clouds, daylight, the moon, etc. But a Fourier transform will still be able to extract the period, even though you may not be able to see it in a simple graph. That is one reason why it is such a powerful technique.
Hi. Thank you very much for your reply!
I think you're right about my misunderstanding. From the science paper I've read by AAVSO, they " performed a Fourier analysis on the resulting combined light curve to obtain the period, ..." , so I assumed Fourier analysis in this case was Fourier series, since I've learned that cepheid variables should undergo pulsations with a regular period, and Fourier series was for functions with the same periodicity. I should redirect my focus and try to learn more about Fourier transform now.

I was trying to plot the AAVSO data on Excel, with the y-axis being the magnitude, and the x-axis being the Julian Date. But I wasn't able to obtain any sort of light-curve-like graph. Could I be misreading the data from the database?
1603546409833.png


A section of the data from AAVSO:
1603546500958.png


Thank you so much again for your help!
 
  • #8
phyzguy
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You are interpreting the data correctly. The point is that it is hard to tell just by looking, because you don't know what the period is. What if the period is shorter than the spacing between your data points? Try taking those data and doing a Fourier transform. Many programs can do a Fourier transform of your data, like Python, Matlab, Maple, Mathematica... After you have an estimate of the period, you can subtract multiples of the period from the JD to get all of the points in a single period. If you include more data from the database, make sure you are only doing data with the same filter band. You don't want to mix B and R data for example. You should do each filter band separately, because there will be offsets in the magnitudes between bands (and even between different observers in the same band). Look at Figure 2 in the paper you linked and try to understand what they mean by "zero-point offsets".
 
  • #9
You are interpreting the data correctly. The point is that it is hard to tell just by looking, because you don't know what the period is. What if the period is shorter than the spacing between your data points? Try taking those data and doing a Fourier transform. Many programs can do a Fourier transform of your data, like Python, Matlab, Maple, Mathematica... After you have an estimate of the period, you can subtract multiples of the period from the JD to get all of the points in a single period. If you include more data from the database, make sure you are only doing data with the same filter band. You don't want to mix B and R data for example. You should do each filter band separately, because there will be offsets in the magnitudes between bands (and even between different observers in the same band). Look at Figure 2 in the paper you linked and try to understand what they mean by "zero-point offsets".
Thanks again for the reply.
Okay, I'll look more into "zero-point offsets."

Actually, I chose this topic for my math IA (for the International Baccalaureate), so from what I understand, I need to be able to demonstrate that I can apply Fourier transform to a set of data points manually (with steps and procedures of the mathematical process). Do you think it's realistic and doable? From papers I've stumbled upon, a lot of them have used coding, as you've mentioned with the different programs.
 
  • #10
phyzguy
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Yes, it's very doable to do it manually, but a little laborious. The algorithm for numerically applying the Fourier Transform to a set of discrete data points is called the "Fast Fourier Transform" or FFT. you will want to Google FFT and understand how it works.
 
  • #11
Yes, it's very doable to do it manually, but a little laborious. The algorithm for numerically applying the Fourier Transform to a set of discrete data points is called the "Fast Fourier Transform" or FFT. you will want to Google FFT and understand how it works.
I just asked my teacher and coding is allowed as long as I show the process/ explain the code. I think I might try to learn more about Fourier transformation and the coding before proceeding with the writing. Thanks so much for your help!!
 
  • #12
jim mcnamara
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@Dr. Courtney has a link to FFT code in his signature. See if that is any help. And in your case FFT seems like a good choice. He may have some comments, too.
 
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  • #13
@Dr. Courtney has a link to FFT code in his signature. See if that is any help. And in your case FFT seems like a good choice. He may have some comments, too.
Will definitely check that out. Thank you!!
 
  • #14
Dr. Courtney
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@Dr. Courtney has a link to FFT code in his signature. See if that is any help. And in your case FFT seems like a good choice. He may have some comments, too.
The link in my signature is to Fourier transform code using explicit integration (EI) rather than the FFT. While working on our paper, "A More Accurate Fourier Transform" (https://arxiv.org/pdf/1507.01832.pdf ), we did download a bunch of data from AAVSO and analyze it with both standard FFT techniques and our EI method.

The bottom line is that Fourier transform methods do OK at recovering the period of many variable stars. But pushing the accuracy envelope in determining period or frequency of a signal really requires the period to be more constant than most variable stars, which only have a constant period to a first order approximation. For this reason, other (non-Fourier) methods are often used. See:

http://spiff.rit.edu/classes/phys445/lectures/period/period.html

http://astro.unl.edu/naap/vsp/pdm.html

Our use of Fourier transforms to analyze variable star data was not particularly accurate or interesting. We ended up leaving it out of our paper. Variable stars is an example of an almost periodic signal that is not particularly amenable to Fourier analysis.
 
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  • #15
phyzguy
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The link in my signature is to Fourier transform code using explicit integration (EI) rather than the FFT. While working on our paper, "A More Accurate Fourier Transform" (https://arxiv.org/pdf/1507.01832.pdf ), we did download a bunch of data from AAVSO and analyze it with both standard FFT techniques and our EI method.

The bottom line is that Fourier transform methods do OK at recovering the period of many variable stars. But pushing the accuracy envelope in determining period or frequency of a signal really requires the period to be more constant than most variable stars, which only have a constant period to a first order approximation. For this reason, other (non-Fourier) methods are often used. See:

http://spiff.rit.edu/classes/phys445/lectures/period/period.html

http://astro.unl.edu/naap/vsp/pdm.html

Our use of Fourier transforms to analyze variable star data was not particularly accurate or interesting. We ended up leaving it out of our paper. Variable stars is an example of an almost periodic signal that is not particularly amenable to Fourier analysis.
While I'm sure what you say is true, I would argue that for a high school student like the OP, he would be better served by first learning well the traditional Fourier transform method before delving into alternate techniques.
 
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  • #16
Dr. Courtney
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While I'm sure what you say is true, I would argue that for a high school student like the OP, he would be better served by first learning well the traditional Fourier transform method before delving into alternate techniques.
Sure, especially since the phrasing of the assignment is telling the student to use Fourier methods.

I'm more pointing out that variable stars are not really the best example of Fourier methods in action.
 
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  • #17
sophiecentaur
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From this paper https://arxiv.org/pdf/1111.0262.pdf
under 'analysis', the period is said to be obtained through fourier series. I didn't quite understand how that was possible
There are several caveats when you try to do this sort of analysis. `````````As already mentioned, if the samples you have are too sparse (below the Nyquist Limit) the data can come up with Alias values which can be lower frequency than the 'real data.
Also, because the data is very truncated, to do any time / frequency transform you can either assume that there is a repeated waveform. The components that the transform yields will all be related to the (arbitrary) choice of length of time that the samples are available for. One way to reduce the error due to this is to apply Windowing.

But there are many misconceptions and over simplifications about this sort of analysis, all of which starts off with the assumption that data exists over all time.

I'm more pointing out that variable stars are not really the best example of Fourier methods in action.
Absolutely. Best to start with long bursts of easily recognisable audio waveforms.
 
  • #18
Yes, it's very doable to do it manually, but a little laborious. The algorithm for numerically applying the Fourier Transform to a set of discrete data points is called the "Fast Fourier Transform" or FFT. you will want to Google FFT and understand how it works.
Hello. Thank you very much for your help! After asking my math teacher, I was able to use code to find the period of the M31_V1 from AAVSO data.
I'm a little stuck on the fourier series aspect in order to obtain the light curve of the graph though.
From this equation, how do I know what the zero-point time is?
1604132360957.png


Does middle time span of observations refer to the averaged time from the AAVSO data?
I.e. finding the mean of the Julian Dates from the 206 raw AAVSO data points (?)
1604132685094.png

...



Also, how do I determine the number of orders? Do I do so experimentally by trial and error?

Which time should I take?

Thank you so much for your help!
 

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  • #20
sophiecentaur
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@no_drama_llama_77 the zero point in time to choose doesn’t matter. You just get a different phase value out of the transform.
 
  • #21
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I was trying to plot the AAVSO data on Excel, with the y-axis being the magnitude, and the x-axis being the Julian Date. But I wasn't able to obtain any sort of light-curve-like graph
It would be a lot more helpful to give us a table of the numbers and not a picture of a table of the numbers.
 
  • #23
@no_drama_llama_77 the zero point in time to choose doesn’t matter. You just get a different phase value out of the transform.
Oh I see, I don't quite understand how to calculate the Fourier coefficients. A lot of the examples online are for a period of 2pi, and assume a range from 0 to 2pi. Is it possible to calculate the Fourier coefficients by substituting a value of t? And what integral range should I use?

I'm a little confused. Thank you
 
  • #24
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I got this from the AAVSO database
Which has all sorts of extra stuff in it.

I was going to whip up some code to help, but not if I have to spend a lot of time struggling with the data.
 
  • #25
Which has all sorts of extra stuff in it.

I was going to whip up some code to help, but not if I have to spend a lot of time struggling with the data.
I organised it into an Excel chart since it was quite hard to read for sure. Sorry about that
 

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