1. Our Solar System: What does it tell us?
2.
Fourier Analysis
i.
Finding periods in your data
ii.
Fitting your data
Earth
Distance: 1.0 AU (1.5 ×1013 cm)
Period: 1 year
Radius: 1 RE (6378 km)
Mass: 1 ME (5.97 ×1027 gm)
Density 5.50 gm/cm3 (densest)
Satellites: Moon (Sodium atmosphere)
Structure: Iron/Nickel Core (~5000 km), rocky mantle
Temperature: -85 to 58 C (mild Greenhouse effect)
Magnetic Field: Modest
Atmosphere: 77% Nitrogen, 21 % Oxygen , CO2, water
Internal Structure of the Earth
Venus
Distance: 0.72 AU
Period: 0.61 years
Radius: 0.94 RE
Mass: 0.82 ME
Density 5.4 gm/cm3
Structure: Similar to Earth Iron Core (~3000 km), rocky mantle
Magnetic Field: None (due to slow rotation)
Atmosphere: Mostly Carbon Dioxide
Internal Structure of Venus
1. Silicate Mantle
Nickel-Iron Core
Crust:
Venus is believed to have an internal structure similar to the Earth
Mars
Distance: 1.5 AU
Period: 1.87 years
Radius: 0.53 RE
Mass: 0.11 ME
Density: 4.0 gm/cm3
Satellites: Phobos and Deimos
Structure: Dense Core (~1700 km), rocky mantle, thin crust
Temperature: -87 to -5 C
Magnetic Field: Weak and variable (some parts strong)
Atmosphere: 95% CO2, 3% Nitrogen, argon, traces of oxygen
Internal Structure of Mars
Mercury
Distance: 0.38 AU
Period: 0.23 years
Radius: 0.38 RE
Mass: 0.055 ME
Density 5.43 gm/cm3 (second densest)
Structure: Iron Core (~1900 km), silicate mantle (~500 km)
Temperature: 90K – 700 K
Magnetic Field: 1% Earth
Internal Structure of Mercury
1. Crust: 100 km
2. Silicate Mantle (25%)
3. Nickel-Iron Core (75%)
Moon
Radius: 0.27 RE
Mass: 0.011 ME
Density: 3.34 gm/cm3
Structure: Dense Core (~1700 km), rocky mantle, thin crust
Internal Structure of the Moon
The moon has a very small core, but a large
mantle (≈70%)
Comparison of Terrestrial Planets
Satellites
R = 0.28 REarth
M = 0.015 MEarth
r = 3.55 gm cm–3
R = 0.25 REarth
M = 0.083MEarth
r = 3.01 gm cm–3
R = 0.41 REarth
M = 0.025MEarth
r = 1.94 gm cm–3
R = 0.38 REarth
M = 0.018 MEarth
r = 1.86 gm cm–3
http://astronomy.nju.edu.cn/~lixd/GA/AT4/AT411/HTML/AT41105.htm
Note: The mean
density increases
with increasing
distance from
Jupiter
Internal Structure of Titan
r (gm/cm3)
10
7
Earth
Mercury
5
4
3
Venus
Mars
Moon
2
From Diana Valencia
1
0.2
0.4
0.6
0.8
1
1.2
Radius (REarth)
1.4
1.6
1.8
2
Jupiter
Distance: 5.2 AU
Period: 11.9 years
Diameter: 11.2 RE (equatorial)
Mass: 318 ME
Density 1.24 gm/cm3
Satellites: > 20
Structure: Rocky Core of 10-13 ME, surrounded by liquid
metallic hydrogen
Temperature: -148 C
Magnetic Field: Huge
Atmosphere: 90% Hydrogen, 10% Helium
From Brian Woodahl
Saturn
Distance: 9.54 AU
Period: 29.47 years
Radius: 9.45 RE (equatorial) = 0.84 RJ
Mass: 95 ME (0.3 MJ)
Density 0.62 gm/cm3 (least dense)
Satellites: > 20
Structure: Similar to Jupiter
Temperature: -178 C
Magnetic Field: Large
Atmosphere: 75% Hydrogen, 25% Helium
Uranus
Distance: 19.2 AU
Period: 84 years
Radius: 4.0 RE (equatorial) = 0.36 RJ
Mass: 14.5 ME (0.05 MJ)
Density: 1.25 gm/cm3
Satellites: > 20
Structure: Rocky Core, Similar to Jupiter but without metallic
hydrogen
Temperature: -216 C
Magnetic Field: Large and decentered
Atmosphere: 85% Hydrogen, 13% Helium, 2% Methane
Neptune
Distance: 30.06 AU
Period: 164 years
Radius: 3.88 RE (equatorial) = 0.35 RJ
Mass: 17 ME (0.05 MJ)
Density: 1.6 gm/cm3 (second densest of giant planets)
Satellites: 7
Structure: Rocky Core, no metallic Hydrogen (like Uranus)
Temperature: -214 C
Magnetic Field: Large
Atmosphere: Hydrogen and Helium
Uranus
Neptune
Comparison of the Giant Planets
1.24
0.62
1.25
1.6
Mean density (gm/cm3)
http://www.freewebs.com/mdreyes3/chaptersix.htm
Neptune
Jupiter
Uranus
Log
Saturn
Jupiter
Saturn
Uranus
Neptune
Venus
CoRoT 7b
Earth
CoRoT 9b
H/He dominated
planets
Ice dominated
planets
Rock/Iron
dominated
planets
Reminder of what a transit curve looks like
II. Fourier Analysis: Searching for Periods in Your Data
Discrete Fourier Transform: Any function can be fit as
a sum of sine and cosines (basis or orthogonal
functions)
N0
FT(w) =  Xj (t) e–iwt
Recall eiwt = cos wt + i sinwt
j=1
X(t) is the time series
1
Power: Px(w) =
| FTX(w)|2
N0
2
1
Px(w) =
Xj cos wtj +
N0
[(S
N0 = number of points
2
) (S X sin wt ) ]
j
j
A DFT gives you as a function of frequency the amplitude
(power = amplitude2) of each sine function that is in the data
Every function can be represented by a sum of sine
(cosine) functions. The FT gives you the amplitude
of these sine (cosine) functions.
FT
P
Ao
Ao
t
1/P
A pure sine wave is a delta function in Fourier space
w
Fourier Transforms
Two important features of Fourier transforms:
1) The “spatial or time coordinate” x maps into a
“frequency” coordinate 1/x (= s or n)
Thus small changes in x map into large changes in s.
A function that is narrow in x is wide in s
The second feature comes later….
A Pictoral Catalog of Fourier Transforms
Time/Space Domain
Time
Fourier/Frequency Domain
0
Frequency (1/time)
Period = 1/frequency
Comb of Shah function
(sampling function)
x
1/x
Time/Space Domain
Fourier/Frequency Domain
Negative
frequencies
Cosine is an even function:
cos(–x) = cos(x)
Positive
frequencies
Time/Space Domain
Sine is an odd function: sin(–x)
= –sin(x)
Fourier/Frequency Domain
Time/Space Domain
Fourier/Frequency Domain
e–px2
w
e–ps2
1/w
The Fourier Transform of a Gausssian is another Gaussian. If the
Gaussian is wide (narrow) in the temporal/spatial domain, it is
narrow(wide) in the Fourier/frequency domain. In the limit of an infinitely
narrow Gaussian (d-function) the Fourier transform is infinitely wide
(constant)
Time/Space Domain
All functions are interchangeable. If
it is a sinc function in time, it is a slit
function in frequency space
Fourier/Frequency Domain
Note: these are the diffraction
patterns of a slit, triangular and
circular apertures
Fourier Transforms : Convolution
Convolution
 f(u)f(x–u)du = f * f
f(x):
f(x):
Fourier Transforms: Convolution
f(x-u)
a2
a1
a3
g(x)
a3
a2
Convolution is a smoothing function
a1
Fourier Transforms
The second important features of Fourier transforms:
2) In Fourier space the convolution is just the product of
the two transforms:
Normal Space
f*g
f g
sinc
Fourier Space
F G
F*G
sinc2
Alias periods:
Undersampled periods appearing as another period
Nyquist Frequency:
The shortest detectable frequency in your
data. If you sample your data at a rate of
Dt, the shortest frequency you can detect
with no aliases is 1/(2Dt)
Example: if you collect photometric data at
the rate of once per night (sampling rate 1
day) you will only be able to detect
frequencies up to 0.5 c/d
In ground based data from one site one
always sees alias frequencies at n + 1
What does a transit light curve look
like in Fourier space?
In time domain
A Fourier transform uses sine function. Can it find a
periodic signal consisting of a transit shape (slit
function)?
n = 0.26 c/d
P = 3.85 d
This is a sync function
caused by the length of
the data window
A short time string of a sine
Sine times step function of length of your
data window
Wide sinc function
d-fnc * step
A longer time string of the same sine
Narrow sinc
function
What happens when you carry out the Fourier
transform of our Transit light curve to higher
frequencies?
The peak of the combs is modulated with a shape
of another sinc function. Why?
In time „space“
* = convolution
=
X
*
Transit shape
Comb
spacing
of P
Length of data string
In frequency „space“
=
*
X
Sinc function of
transit shape
Comb
spacing of
1/P
Sinc of data
window
But wait, the observed light curve is not a continuous function.
One should multiply by a comb function of your sampling rate.
Thus this observed transform should be convolved with another
comb.
Frequencies
repeat
This pattern gets
repeated in intervals of
200 c/d for this
sampling. Frequencies
on either side of the
peak are –n and +n
Nyquist
When you go to higher frequencies you see this. In this case the
sampling rate is 0.005 d, thus the the pattern is repeated on a comb
every 200 c/d. Frequencies at the Nyquist frequency of 100 d.
One generally does not compute the FT for frequencies beyond the
Nyquist frequencies since these repeat and are aliases.
t = 0.125 d
1/t
The duration of the transit is related to the
location of the first zero in the sinc function
that modulates the entire Fourier transform
In principle one can use the Fourier transform of
your light curve to get the transit period and
transit duration. What limits you from doing this is
the sampling window and noise.
The effects of noise in your data
Little noise
Signal level
More noise
A lot of
noise
Noise level
Transit period of 3.85 d
(frequency = 0.26 c/d)
The Effects of Sampling
This is the previous transit light curve with
more realistic sampling typical of what you
can achieve from the ground.
20 d?
20 d
Frequency (c/d)
Time (d)
Sampling creates aliases and spectral leakage which
produces „false peaks“ that make it difficult to chose the
correct period that is in the data.
A very nice sine fit to data….
P = 3.16 d
That was generated with pure random noise and no signal
After you have found a periodic signal in your data you must ask
yourself „What is the probability that noise would also produce this
signal? This is commonly called the False Alarm Probability (FAP)
1. Is there a periodic
signal in my data?
no
yes
Stop
A Flow Diagram
for making exciting
discoveries
2. Is it due to Noise?
yes
Stop
no
3. What is its Nature?
4. Is this interesting?
no
yes
5. Publish results
Find another star
Period Analysis with Lomb-Scargle Periodograms
LS Periodograms are useful for assessing the statistical
signficance of a signal
1
Px(w) =
2
[ S X cos w(t –t)]
j
j
S
2
j
1
+
2
2
Xj cos w(tj–t)
j
tan(2wt) =
[ S X sin w(t –t) ]
j
j
j
S X sin
j
2
w(tj–t)
(Ssin
2wtj)/(Scos 2wtj)
j
j
In a normal Fourier Transform the Amplitude (or Power) of a
frequency is just the amplitude of that sine wave that is present
in the data.
In a Scargle Periodogram the power is a measure of the
statistical significance of that frequency (i.e. is the signal real?)
2
Scargle Periodogram
Amplitude (m/s)
Fourier Transform
Note: Square this for a direct
comparison to Scargle: power to
power
FT and Scargle have different „Power“ units
Period Analysis with Lomb-Scargle Periodograms
If P is the „Scargle Power“ of a peak in the Scargle periodogram we
have two cases to consider:
1. You are looking for an unknown period. In this case you must
ask „What is the FAP that random noise will produce a peak
higher than the peak in your data periodogram over a certain
frequency interval n1 < n < n2. This is given by:
False alarm probability ≈ 1 – (1–e–P)N ≈ Ne–P
N = number of indepedent frequencies ≈ number of data points
Horne & Baliunas (1986), Astrophysical Journal, 302, 757 found an
empirical relationship between the number of independent frequencies,
Ni, and the number of data points, N0 :
Ni = –6.362 + 1.193 N0 + 0.00098 N02
Example: Suppose you have 40 measurements
of a star that has periodic variations and you find
a peak in the periodogram. The Scargle power,
P, would have to have a value of ≈ 8.3 for the
FAP to be 0.01 ( a 1% chance that it is noise).
2. There is a known period (frequency) in your data. This is
often the case in transit work where you have a known
photometric period, but you are looking for the same period
in your radial velocity data. You are now asking „What is the
probability that noise will produce a peak exactly at this
frequency that has more power than the peak found in the
data?“ In this case the number of independent frequencies is
just one: N = 1. The FAP now becomes:
False alarm probability = e–P
Example: Regardless of how many measurements you have
the Scargle power should be greater than about 4.6 to have a
FAP of 0.01 for a known period (frequency)
Fourier Amplitude
Noisy data
Less Noisy
data
In a normal Fourier
transform the Amplitude
of a peak stays the
same, but the noise level
drops
versus Lomb-Scargle Amplitude
In a Scargle periodogram the noise level drops, but the power in the
peak increases to reflect the higher significance of the detection.
Two ways to increase the significance: 1) Take better data (less noise)
or 2) Take more observations (more data). In this figure the red curve is
the Scargle periodogram of transit data with the same noise level as
the blue curve, but with more data measurements.
Assessing the False Alarm Probability: Random Data
The best way to assess the FAP is through Monte Carlo
simulations:
Method 1: Create random noise with the same standard
deviation, s, as your data. Sample it in the same way as the
data. Calculate the periodogram and see if there is a peak
with power higher than in your data over a speficied
frequency range. If you are fitting sine wave see if you have a
lower c2 for the best fitting sine wave. Do this a large number
of times (1000-100000). The number of periodograms with
power larger than in your data, or c2 for sine fitting that is
lower gives you the FAP.
Assessing the False Alarm Probability: Bootstrap Method
Method 2: Method 1 assumes that your noise distribution is
Gaussian. What if it is not? Then randomly shuffle your actual
data values keeping the times fixed. Calculate the
periodogram and see if there is a peak with power higher than
in your data over a specified frequency range. If you are fitting
sine wave see if you have a lower c2 for the best fitting sine
function. Shuffle your data a large number of times (1000100000). The number of periodograms in your shuffled data
with power larger than in your data, or c2 for sine fitting that
are lower gives you the FAP.
This is my preferred method as it preserves the noise
characteristics in your data. It is also a conservative
estimate because if you have a true signal your shuffling
is also including signal rather than noise (i.e. your noise is
lower)
Least Squares Sine Fitting
Fit a sine wave of the form:
y(t) = A·sin(wt + f) + Constant
Where w = 2p/P, f = phase shift
Best fit minimizes the c2:
c2 = S (di –gi)2/N
di = data, gi = fit
Sine fitting is more appropriate if you have
few data points. Scargle estimates the
noise from the rms scatter of the data
regardless if a signal is present in your
data. The peak in the periodogram will thus
have a lower significance even if there is
really a signal in the data. But beware, one
can find lots of good sine fits to noise!
Most algorithms (fortran and c language) can be found in
Numerical Recipes
Period04: multi-sine fitting with Fourier analysis.
Tutorials available plus versions in Mac OS, Windows,
and Linux
http://www.univie.ac.at/tops/Period04/
The first Tautenburg Planet: HD 13189
Amplitude (m/s)
Least squares sine fitting: The best
fit period (frequency) has the
lowest c2
Discrete Fourier Transform: Gives
the power of each frequency that is
present in the data. Power is in
(m/s)2 or (m/s) for amplitude
Lomb-Scargle Periodogram: Gives
the power of each frequency that is
present in the data. Power is a
measure of statistical signficance
Fourier Analysis: Removing unwanted signals
Sines and Cosines form a basis. This means that every
function can be modeled as a infinite series of sines and
cosines. This is useful for fitting time series data and
removing unwanted signals.
Example. For a function y = x over the interval x = 0,L
you can calculate the Fourier coefficients and get that the
amplitudes of the sine waves are
Bn = (–1) n+1 (2kL/np)
Fitting a step functions with sines
See file corot2b.dat for light curve
See file corot7b.dat and corot7b.p04
Prot = 23 d
0.035%
PTransit = 0.85 d = 1.176 d