Fourier Series and Cepheid Variables

In summary, the conversation discusses using Fourier series to find the period of a cepheid variable from a set of data points for its magnitude at different times. The conversation includes confusion about the difference between Fourier series and Fourier transform, as well as attempts to plot the data and obtain a light curve. It is suggested to use a Fourier transform to extract the period from the data and to be cautious when using data from different filter bands.
  • #1
no_drama_llama_77
26
2
TL;DR 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!
 
Astronomy news on Phys.org
  • #2
To get the period you would data at a series of times. I don't understand what you have in mind.
 
  • #3
mathman said:
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
mathman said:
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
The data points are magnitudes at many times. The fluctuations give the period.
 
  • #6
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.
 
  • Like
  • Informative
Likes jim mcnamara and no_drama_llama_77
  • #7
phyzguy said:
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
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
phyzguy said:
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
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
phyzguy said:
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
@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.
 
  • Like
Likes no_drama_llama_77
  • #13
jim mcnamara said:
@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
jim mcnamara said:
@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.
 
  • Like
Likes jim mcnamara
  • #15
Dr. Courtney said:
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.
 
  • Like
Likes sophiecentaur and no_drama_llama_77
  • #16
phyzguy said:
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.
 
  • Like
Likes sophiecentaur and phyzguy
  • #17
no_drama_llama_77 said:
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.

Dr. Courtney said:
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
phyzguy said:
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!
 

Attachments

  • 1604132633155.png
    1604132633155.png
    33.9 KB · Views: 209
  • 1604132640696.png
    1604132640696.png
    3.6 KB · Views: 208
Last edited:
  • #20
@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
no_drama_llama_77 said:
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.
 
  • #22
Vanadium 50 said:
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.

I got this from the AAVSO database
 

Attachments

  • aavsodata_5f913ba243808.txt
    86.5 KB · Views: 160
  • #23
sophiecentaur said:
@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
no_drama_llama_77 said:
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
Vanadium 50 said:
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
 

Attachments

  • M31_V1 data.xlsx
    21.1 KB · Views: 219
  • #26
no_drama_llama_77 said:
I organised it into an Excel chart since it was quite hard to read for sure.

Actually, I found a code to find the period (using Lomb Scargle Periodogram), however, I'm struggling with how to mathematically work out the Fourier Series for the light curve based on the two frequency peaks I found from my power- frequency graph, assuming the peak at 0 frequency is an error/noise(?). (I've attached my spectral density graph obtained from the coding)
Since this is for my math IA for the IB (international baccalaureate), I need to show some mathematical calculations.

Thank you for your help.
 

Attachments

  • Spectral Density Graph.png
    Spectral Density Graph.png
    5.3 KB · Views: 157
  • #27
I looked at the Excel data, du0plicated your chart, and am wondering if the data has problems.

First oddity: the pre-1927 data looks different than the post-1950 data.
Second oddity: the gap between 1927 and 1950
Third oddity: 2423610.6 has two values, not very close to each other

If I take the data seriously, the change in magnitude per day is around 0.2 or 0.3. I'm looking at a swing of maybe 1.2, so I am looking at a period of maybe 4 days. But many of your samples are a month apart.

A Fourier Transform can be thought of as a clever way to fit a sine wave to your data. It will have trouble here because the data is comprised of short runs, much less than a period, separated by long runs, much more than a period. This is the hardest case, and only works when the frequency is very sharp and the shape is very stable.

That's not the case we have.
 
  • #28
Vanadium 50 said:
I looked at the Excel data, du0plicated your chart, and am wondering if the data has problems.

First oddity: the pre-1927 data looks different than the post-1950 data.
Second oddity: the gap between 1927 and 1950
Third oddity: 2423610.6 has two values, not very close to each other

If I take the data seriously, the change in magnitude per day is around 0.2 or 0.3. I'm looking at a swing of maybe 1.2, so I am looking at a period of maybe 4 days. But many of your samples are a month apart.

A Fourier Transform can be thought of as a clever way to fit a sine wave to your data. It will have trouble here because the data is comprised of short runs, much less than a period, separated by long runs, much more than a period. This is the hardest case, and only works when the frequency is very sharp and the shape is very stable.

That's not the case we have.

Thanks for taking the time to help me.

I ran a Lomb Scargle Periodogram analysis on the raw data instead of a normal Fourier Transform for the same reasons you've mentioned: the data is quite unevenly spaced. As for the 1st and 3rd oddity, I didn't think much about it as I'm very new to processing this kind of data, but as a pointer given by @phyzguy , I made sure to only process the data that was collected with the same filter band (in this case B). So I'm assuming the uncertainty should be roughly the same as the same filter band was being used?
 
  • #29
no_drama_llama_77 said:
I ran a Lomb Scargle Periodogram analysis

That is almost certainly just noise.

Your minimum time between samples is 0.1 days. (I suspect that it's even worse and it's just rounded to 0.1 days, but let's assume it's 0.1 days). Thats' 8640 seconds, so the highest frequency you can possibly see is 0.00012 Hz. That's all contained in your first "peak". All the peaks to the right are pure noise and artifacts.
 
  • #30
Vanadium 50 said:
I looked at the Excel data, du0plicated your chart, and am wondering if the data has problems.

First oddity: the pre-1927 data looks different than the post-1950 data.
Second oddity: the gap between 1927 and 1950
Third oddity: 2423610.6 has two values, not very close to each other

If I take the data seriously, the change in magnitude per day is around 0.2 or 0.3. I'm looking at a swing of maybe 1.2, so I am looking at a period of maybe 4 days. But many of your samples are a month apart.

A Fourier Transform can be thought of as a clever way to fit a sine wave to your data. It will have trouble here because the data is comprised of short runs, much less than a period, separated by long runs, much more than a period. This is the hardest case, and only works when the frequency is very sharp and the shape is very stable.

That's not the case we have.

Thanks for taking the time to help me.

I ran a Lomb Scargle Periodogram analysis on the raw data, for the same reasons you've mentioned: the data is quite unevenly spaced. As for the 1st and 3rd oddity, I didn't think much about it as I'm not a
Vanadium 50 said:
That is almost certainly just noise.

Your minimum time between samples is 0.1 days. (I suspect that it's even worse and it's just rounded to 0.1 days, but let's assume it's 0.1 days). Thats' 8640 seconds, so the highest frequency you can possibly see is 0.00012 Hz. That's all contained in your first "peak". All the peaks to the right are pure noise and artifacts.
Actually, I'm assuming the period is one out of the two second-highest peaks, one on the left and the other on the right.
From my calculations
Period for the left frequency: 502.3156753… days
Period for the right frequency: 31.38948703… days

A paper has made observations on the period of M31_V1, and they obtained a period of 31.4 days. Therefore, I think the peak on the right is not noise but rather the closest to the correct period, while the frequency peaks on the left are noise or an alias period?
https://arxiv.org/pdf/1111.0262.pdf

Anyway, I think I'm okay with the first part of finding the period, but I really need help with Fourier Series analysis (as per my previous posts). If possible can you please help me with trying to understand how to manually work out the Fourier series function for the supposed light curve for M31_V1?

Thank you.
 
  • #31
no_drama_llama_77 said:
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?

UPDATE:
In dire need of Fourier Series help.
If anyone can please help me with understanding how to calculate the Fourier series based on the AAVSO raw data and the periods I've obtained from the spectral density graph gotten from coding, I will be very grateful.
(Read posts above)

Thank you so much T-T
 
  • #32
no_drama_llama_77 said:
From my calculations
Period for the left frequency: 502.3156753… days
Period for the right frequency: 31.38948703… days

A. Please watch the precision. You are not getting 11 digits here no matter what you do: that's 10 ms over the entire observing period.
B. Please wtach your units: the tick marks go from 0.005 Hz to 0.030 Hz. That's a period from 30 to 200 seconds. You can't see a 30 day period on this plot, nor distinguish it from 500 days.

If you sample every 2.4 hours, you can't see a 200 second effect. It's got to be noise.
 
  • #33
Vanadium 50 said:
A. Please watch the precision. You are not getting 11 digits here no matter what you do: that's 10 ms over the entire observing period.
B. Please wtach your units: the tick marks go from 0.005 Hz to 0.030 Hz. That's a period from 30 to 200 seconds. You can't see a 30 day period on this plot, nor distinguish it from 500 days.

If you sample every 2.4 hours, you can't see a 200 second effect. It's got to be noise.

A. I wasn't sure what the data uncertainty was, so I rounded my figures to 3 significant figures in my written report. Will that be appropriate?

B. When I apply the Lomb Scargle Periodogram to the data, does that mean my frequency is actually represented by the units (1/days) rather than the usual (1/seconds), since the AAVSO data measures the period using Julian dates (days)? I may have mislabelled my spectral density graph.
I obtained 502 frequency by dividing 1/0.00199078. I obtained 0.00199078 by estimating the coordinates of the second-highest peak on the left

1604164262920.png
 
  • #34
no_drama_llama_77 said:
A. I wasn't sure what the data uncertainty was, so I rounded my figures to 3 significant figures in my written report. Will that be appropriate?

Do you think so? That means you think you know the period to one part in a thousand. (And for that matter, that the period is stable to one part in a thousand)

no_drama_llama_77 said:
When I apply the Lomb Scargle Periodogram to the data, does that mean my frequency is actually represented by the units (1/days) rather than the usual (1/seconds), since the AAVSO data measures the period using Julian dates (days)

If days are the input than 1/days is the output.

That said, I am still far from convinced that you are doing anything besides chasing fluctuations.
A. If you split the data into pre-1940 and post-1940 do you get the same answer?
B. If you fit even and odd data points do you get the same answer?
 
  • #35
no_drama_llama_77 said:
A paper has made observations on the period of M31_V1, and they obtained a period of 31.4 days. Therefore, I think the peak on the right is not noise but rather the closest to the correct period

"I got the same result as someone else, and therefore I must be right" is not the way we do science. People have gotten burned doing this.
 
<h2>1. What is a Fourier series?</h2><p>A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It is named after French mathematician Joseph Fourier and is commonly used in signal processing, physics, and engineering.</p><h2>2. How are Fourier series and Cepheid variables related?</h2><p>Cepheid variables are a type of star that exhibits periodic changes in brightness. These changes can be described by a Fourier series, which helps astronomers study and understand the behavior of Cepheid variables.</p><h2>3. Why are Cepheid variables important in astronomy?</h2><p>Cepheid variables are important because they are used as standard candles to measure distances in the universe. By studying their variations in brightness, astronomers can determine their intrinsic luminosity and use this to calculate their distance from Earth.</p><h2>4. How do scientists use Fourier series to study Cepheid variables?</h2><p>Scientists use Fourier series to analyze the light curves of Cepheid variables, which show how their brightness changes over time. By fitting a Fourier series to the light curve, they can determine the period, amplitude, and shape of the variations, which can provide valuable information about the star's physical properties.</p><h2>5. Can Fourier series be used to study other types of variable stars?</h2><p>Yes, Fourier series can be used to study many different types of variable stars, including eclipsing binaries, RR Lyrae stars, and pulsating white dwarfs. By analyzing their light curves with Fourier series, scientists can learn more about the physical processes and properties of these stars.</p>

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It is named after French mathematician Joseph Fourier and is commonly used in signal processing, physics, and engineering.

2. How are Fourier series and Cepheid variables related?

Cepheid variables are a type of star that exhibits periodic changes in brightness. These changes can be described by a Fourier series, which helps astronomers study and understand the behavior of Cepheid variables.

3. Why are Cepheid variables important in astronomy?

Cepheid variables are important because they are used as standard candles to measure distances in the universe. By studying their variations in brightness, astronomers can determine their intrinsic luminosity and use this to calculate their distance from Earth.

4. How do scientists use Fourier series to study Cepheid variables?

Scientists use Fourier series to analyze the light curves of Cepheid variables, which show how their brightness changes over time. By fitting a Fourier series to the light curve, they can determine the period, amplitude, and shape of the variations, which can provide valuable information about the star's physical properties.

5. Can Fourier series be used to study other types of variable stars?

Yes, Fourier series can be used to study many different types of variable stars, including eclipsing binaries, RR Lyrae stars, and pulsating white dwarfs. By analyzing their light curves with Fourier series, scientists can learn more about the physical processes and properties of these stars.

Similar threads

  • Science and Math Textbooks
Replies
6
Views
1K
  • Calculus
Replies
8
Views
4K
Replies
11
Views
806
  • Topology and Analysis
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
993
  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Classical Physics
Replies
3
Views
2K
Replies
4
Views
1K
  • Astronomy and Astrophysics
Replies
1
Views
2K
Back
Top