![Fourier Transforms and Galactic Spectroscopy
Jared Cohn Michigan State University PHY415
Introduction to Fourier Transforms
A Fourier transform is closely related to a complex Fourier series, but allowing for a
continuous function rather than a periodic one. Whereas the complex Fourier series
is summed such that n is an integer divided by two times the period, the transform
assumes that the function is continuous from negative to positive infinite, and can
thus be summed as an integral.
Complex Fourier series are calculated through discrete sums of complex
exponentials, and those coefficients can be defined as sampled values of a
corresponding Fourier transform.
The transform can accept raw data, of a non-periodic form, and turn it into a
frequency distribution, which can then be easily interpreted.
The Fourier series, which is represented as a sum of sines and cosines can be
dissected into its component functions. The frequency with which these functions
occur is represented by the Fourier transform. In this way, the Fourier transform can
measure an intensity of an incident function, transform it and produce a spectrum of
its components.
How is the transform related to the series?
𝑎0 =
1
2𝐿
𝑓 𝑥 𝑑𝑥
𝐿
−𝐿
𝑎𝑛𝑑 𝑎 𝑛 =
1
𝐿
𝑓 𝑥 cos
𝑛𝜋𝑥
𝐿
𝑑𝑥
𝐿
−𝐿
𝑏 𝑛 =
1
𝐿
𝑓 𝑥 sin
𝑛𝜋𝑥
𝐿
𝑑𝑥
𝐿
−𝐿
substitute: 𝛼 =
𝑛𝜋
𝐿
Using trigonometric identities:
cos 𝛼 𝑥 − 𝑡 = cos 𝛼𝑥 cos 𝛼𝑡 + sin 𝛼𝑥 sin 𝛼𝑡
The series then reduces to:
𝑓 𝑥 =
1
2𝐿
𝑓 𝑥 𝑑𝑥
𝐿
−𝐿
+
1
2𝜋
𝑓 𝑥 cos 𝛼 𝑥 − 𝑡 𝑑𝑥
𝐿
−𝐿
∞
𝑛=1
Since 𝛼 𝑛 = 𝑛𝜋/𝐿 and 𝛼−𝑛 = −
𝑛𝜋
𝐿
we can say −𝛼 𝑛= 𝛼−𝑛 and can then use this to
substitute in Euler’s equation.
𝑓 𝑥 =
1
2𝐿
𝑓 𝑥 𝑒 𝑖𝛼(𝑥−𝑡)
𝑑𝑥
𝐿
−𝐿
∞
−∞
Now, define 𝑗 ≡
𝜋
𝐿
and substitute it in:
𝑓 𝑥 = 1/(2𝜋) 𝑗 𝑓 𝑥 𝑒−𝑛𝑗(𝑥−𝑡)
𝑑𝑥
𝐿
−𝐿
∞
−∞
And let j go to 0 and L tends toward infinite. Leaving a function in only alpha and x.
We will call it F(α).
𝑓 𝑥 =
1
2𝜋
𝐹 𝛼 𝑑𝛼
∞
−∞
Which becomes the inverse Fourier transform:
𝑓 𝑥 =
1
2𝜋
𝑔 𝛼 𝑒−𝑖𝛼𝑥
∞
−∞
And finally the Fourier transform itself:
𝑔 𝛼 = 𝑓 𝑥 𝑒 𝑖𝛼𝑥
𝑑𝑥
∞
−∞
About Fourier Transform Spectroscopy
Spectroscopy is the science of taking light of various wavelengths and using it to determine the makeup
of what is being viewed. Many chemicals and elements have well known spectroscopic lines, and as
such can serve as a basis for making further measurements. Fourier spectroscopy is frequently
performed using a Michelson interferometer with a sliding mirror. This movable mirror allows us to
create interference at various wavelengths, giving us scattered data of many wavelengths at a time. This
raw data creates an interferogram, which can be plugged into a computer that converts the data, via a
Fourier transform, to a readable spectrum. For our purposes, this is frequently done with radio
emissions and infra-red sources, since they are not visible to the human eye.
The spectrum is extracted from the interferogram in the following manner:
The intensity of the light collected by the interferometer is defined as:
𝐼 𝑝, 𝑣 = (𝐼(𝑣)[1 + cos 2𝜋𝑣 𝑝 ]
In which p is the optical path length difference in the interferometer, and 𝑣 is the wave number.
The total intensity over a given path length difference can therefore be represented by an integral:
𝐼 𝑝 = 𝐼 𝑝, 𝑣 = 𝐼 𝑣 [1 + cos 2𝜋𝑣 𝑝 ] ∙ 𝑑𝑣
∞
0
The presence of this cosine indicates are cosine Fourier transform relationship, and so this data can be
turned into a spectrum by a Fourier transform:
𝐼 𝑣 = 4 [𝐼(
∞
0
𝑝) −
1
2
𝐼 𝑝 = 0 cos(2𝜋𝑣 𝑝) ∙ 𝑑𝑝
This produces a readable spectrum.
Galactic Rotation Curves
Galactic rotation curves show us the angular velocities at radial distances from the center of a galaxy.
This is generally done by comparing a well known and very common spectroscopic line (the hydrogen-α)
line as measured in the spectrum to the same line as measured in a laboratory. Due to the Doppler
effect, if we are facing a galaxy such that one side appears to be rotating away from us and the other to
be rotating toward us, we will find that the wavelengths from the hydrogen-α line is distorted by these
velocities. If the galaxy is rotating away from us, the wavelength will appear longer. This is what’s known
as red shift. If it is rotating toward us, the wavelength will appear shorter, known as blue shift.
Kepler’s Laws and Newtonian Orbits
According to the observations of Johannes Kepler, frequency of orbits were proportional to
their distance from the center of mass in the solar system. Newton expanded this idea into
what became his universal theory of gravity. Newton’s theory was able to accurately predict
everything we observed for hundreds of years. Our observations before the mid-20th
century supported that an observable mass would be responsible for detectable gravitation,
and it still holds [accurately] for objects of a sub-galactic scale.
Experiments
In the 1960’s, Vera Rubin began to study galactic rotation curves, and began to find
a discrepancy between rotation curves calculated through Newtonian mechanics, in
which the mass of the galaxy was measured proportional to the luminosity, and the
actual measured values. This was done by comparing the frequency at which light
was observed at various wavelengths.
http://www.astro-photography.net/images/NGC%206503%20rotation%20curve%20annotated.jpg
http://ircamera.as.arizona.edu/NatSci102/NatSci102/images/rot21cm.jpg
Prior to the 1970’s, rotation curves were expected to follow the curve
represented by only the luminous matter line. Since so many conflicting data
points were found, this model had to be revised.
Results
These results indicated that a new model was necessary, since the rotation of the
galaxy should drop off substantially with the square of the distance from the
center, yet the speed was found to level off and be relatively constant much
further away from the galactic center. These results do not prove Newton wrong,
but rather just require that additional matter be present. Since our
measurements of matter are taken via luminosity, and do not seem to fit, there
must be some unobservable matter spread throughout the galaxy. Dark matter is
the name we have chosen for the unobservable mass.
After measuring innumerable galaxies curves, there has been found a frequent
problem of ‘missing mass’.
http://utahscience.oremjr.alpine.k12.ut.us/sciber06/9th/stand_1/images/H2shift.jpg
Works Cited
• Panek, Richard The 4% Universe. Mariner Books, 2011. Print
• “Fourier Transforms: Derivation” JPOffline.com
http://www.jpoffline.com/physics_docs/y2s4/cvit_ft_derivation.pdf
• Boas, Mary L. Mathematical Methods in the Physical Sciences. “Fourier
Transforms”, pp.378-383. (2006). Print.
http://chemwiki.ucdavis.edu/@api/deki/files/9223/Figure_2.png](https://image.slidesharecdn.com/081ea4fe-104f-452f-895e-bcbbf10918eb-161213021946/85/Galactic-Rotation-1-320.jpg)
Galactic Rotation
- 1. Fourier Transforms and Galactic Spectroscopy Jared Cohn Michigan State University PHY415 Introduction to Fourier Transforms A Fourier transform is closely related to a complex Fourier series, but allowing for a continuous function rather than a periodic one. Whereas the complex Fourier series is summed such that n is an integer divided by two times the period, the transform assumes that the function is continuous from negative to positive infinite, and can thus be summed as an integral. Complex Fourier series are calculated through discrete sums of complex exponentials, and those coefficients can be defined as sampled values of a corresponding Fourier transform. The transform can accept raw data, of a non-periodic form, and turn it into a frequency distribution, which can then be easily interpreted. The Fourier series, which is represented as a sum of sines and cosines can be dissected into its component functions. The frequency with which these functions occur is represented by the Fourier transform. In this way, the Fourier transform can measure an intensity of an incident function, transform it and produce a spectrum of its components. How is the transform related to the series? 𝑎0 = 1 2𝐿 𝑓 𝑥 𝑑𝑥 𝐿 −𝐿 𝑎𝑛𝑑 𝑎 𝑛 = 1 𝐿 𝑓 𝑥 cos 𝑛𝜋𝑥 𝐿 𝑑𝑥 𝐿 −𝐿 𝑏 𝑛 = 1 𝐿 𝑓 𝑥 sin 𝑛𝜋𝑥 𝐿 𝑑𝑥 𝐿 −𝐿 substitute: 𝛼 = 𝑛𝜋 𝐿 Using trigonometric identities: cos 𝛼 𝑥 − 𝑡 = cos 𝛼𝑥 cos 𝛼𝑡 + sin 𝛼𝑥 sin 𝛼𝑡 The series then reduces to: 𝑓 𝑥 = 1 2𝐿 𝑓 𝑥 𝑑𝑥 𝐿 −𝐿 + 1 2𝜋 𝑓 𝑥 cos 𝛼 𝑥 − 𝑡 𝑑𝑥 𝐿 −𝐿 ∞ 𝑛=1 Since 𝛼 𝑛 = 𝑛𝜋/𝐿 and 𝛼−𝑛 = − 𝑛𝜋 𝐿 we can say −𝛼 𝑛= 𝛼−𝑛 and can then use this to substitute in Euler’s equation. 𝑓 𝑥 = 1 2𝐿 𝑓 𝑥 𝑒 𝑖𝛼(𝑥−𝑡) 𝑑𝑥 𝐿 −𝐿 ∞ −∞ Now, define 𝑗 ≡ 𝜋 𝐿 and substitute it in: 𝑓 𝑥 = 1/(2𝜋) 𝑗 𝑓 𝑥 𝑒−𝑛𝑗(𝑥−𝑡) 𝑑𝑥 𝐿 −𝐿 ∞ −∞ And let j go to 0 and L tends toward infinite. Leaving a function in only alpha and x. We will call it F(α). 𝑓 𝑥 = 1 2𝜋 𝐹 𝛼 𝑑𝛼 ∞ −∞ Which becomes the inverse Fourier transform: 𝑓 𝑥 = 1 2𝜋 𝑔 𝛼 𝑒−𝑖𝛼𝑥 ∞ −∞ And finally the Fourier transform itself: 𝑔 𝛼 = 𝑓 𝑥 𝑒 𝑖𝛼𝑥 𝑑𝑥 ∞ −∞ About Fourier Transform Spectroscopy Spectroscopy is the science of taking light of various wavelengths and using it to determine the makeup of what is being viewed. Many chemicals and elements have well known spectroscopic lines, and as such can serve as a basis for making further measurements. Fourier spectroscopy is frequently performed using a Michelson interferometer with a sliding mirror. This movable mirror allows us to create interference at various wavelengths, giving us scattered data of many wavelengths at a time. This raw data creates an interferogram, which can be plugged into a computer that converts the data, via a Fourier transform, to a readable spectrum. For our purposes, this is frequently done with radio emissions and infra-red sources, since they are not visible to the human eye. The spectrum is extracted from the interferogram in the following manner: The intensity of the light collected by the interferometer is defined as: 𝐼 𝑝, 𝑣 = (𝐼(𝑣)[1 + cos 2𝜋𝑣 𝑝 ] In which p is the optical path length difference in the interferometer, and 𝑣 is the wave number. The total intensity over a given path length difference can therefore be represented by an integral: 𝐼 𝑝 = 𝐼 𝑝, 𝑣 = 𝐼 𝑣 [1 + cos 2𝜋𝑣 𝑝 ] ∙ 𝑑𝑣 ∞ 0 The presence of this cosine indicates are cosine Fourier transform relationship, and so this data can be turned into a spectrum by a Fourier transform: 𝐼 𝑣 = 4 [𝐼( ∞ 0 𝑝) − 1 2 𝐼 𝑝 = 0 cos(2𝜋𝑣 𝑝) ∙ 𝑑𝑝 This produces a readable spectrum. Galactic Rotation Curves Galactic rotation curves show us the angular velocities at radial distances from the center of a galaxy. This is generally done by comparing a well known and very common spectroscopic line (the hydrogen-α) line as measured in the spectrum to the same line as measured in a laboratory. Due to the Doppler effect, if we are facing a galaxy such that one side appears to be rotating away from us and the other to be rotating toward us, we will find that the wavelengths from the hydrogen-α line is distorted by these velocities. If the galaxy is rotating away from us, the wavelength will appear longer. This is what’s known as red shift. If it is rotating toward us, the wavelength will appear shorter, known as blue shift. Kepler’s Laws and Newtonian Orbits According to the observations of Johannes Kepler, frequency of orbits were proportional to their distance from the center of mass in the solar system. Newton expanded this idea into what became his universal theory of gravity. Newton’s theory was able to accurately predict everything we observed for hundreds of years. Our observations before the mid-20th century supported that an observable mass would be responsible for detectable gravitation, and it still holds [accurately] for objects of a sub-galactic scale. Experiments In the 1960’s, Vera Rubin began to study galactic rotation curves, and began to find a discrepancy between rotation curves calculated through Newtonian mechanics, in which the mass of the galaxy was measured proportional to the luminosity, and the actual measured values. This was done by comparing the frequency at which light was observed at various wavelengths. http://www.astro-photography.net/images/NGC%206503%20rotation%20curve%20annotated.jpg http://ircamera.as.arizona.edu/NatSci102/NatSci102/images/rot21cm.jpg Prior to the 1970’s, rotation curves were expected to follow the curve represented by only the luminous matter line. Since so many conflicting data points were found, this model had to be revised. Results These results indicated that a new model was necessary, since the rotation of the galaxy should drop off substantially with the square of the distance from the center, yet the speed was found to level off and be relatively constant much further away from the galactic center. These results do not prove Newton wrong, but rather just require that additional matter be present. Since our measurements of matter are taken via luminosity, and do not seem to fit, there must be some unobservable matter spread throughout the galaxy. Dark matter is the name we have chosen for the unobservable mass. After measuring innumerable galaxies curves, there has been found a frequent problem of ‘missing mass’. http://utahscience.oremjr.alpine.k12.ut.us/sciber06/9th/stand_1/images/H2shift.jpg Works Cited • Panek, Richard The 4% Universe. Mariner Books, 2011. Print • “Fourier Transforms: Derivation” JPOffline.com http://www.jpoffline.com/physics_docs/y2s4/cvit_ft_derivation.pdf • Boas, Mary L. Mathematical Methods in the Physical Sciences. “Fourier Transforms”, pp.378-383. (2006). Print. http://chemwiki.ucdavis.edu/@api/deki/files/9223/Figure_2.png