PowerPoint-Pr

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• Our Solar System What does it tell us?
• Fourier Analysis
• Finding periods in your data
• Fitting your data

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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
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Internal Structure of the Earth
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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
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Internal Structure of Venus

1. Silicate Mantle

Nickel-Iron Core
Crust
Venus is believed to have an internal structure
similar to the Earth
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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
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Internal Structure of Mars
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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
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Internal Structure of Mercury

1. Crust 100 km
2. Silicate Mantle (25)
3. Nickel-Iron Core (75)

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Moon
Radius 0.27 RE Mass 0.011 ME Density 3.34
gm/cm3 Structure Dense Core (1700 km), rocky
mantle, thin crust
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Internal Structure of the Moon
The moon has a very small core, but a large
mantle (70)
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Comparison of Terrestrial Planets
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Satellites
R 0.28 REarth M 0.015 MEarth r 3.55 gm cm3
Note The mean density increases with increasing
distance from Jupiter
R 0.25 REarth M 0.083MEarth r 3.01 gm cm3
R 0.41 REarth M 0.025MEarth r 1.94 gm cm3
R 0.38 REarth M 0.018 MEarth r 1.86 gm cm3
http//astronomy.nju.edu.cn/lixd/GA/AT4/AT411/HTM
L/AT41105.htm
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Internal Structure of Titan
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Iron enriched
Earth-like
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Earth
Mercury
r (gm/cm3)
No iron
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Venus
Mars
4
3
Moon
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From Diana Valencia
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2
1.8
0.2
0.4
1
0.6
0.8
1.2
1.4
1.6
Radius (REarth)
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Jupiter
Distance 5.2 AU Period 11.9 years Diameter
11.2 RE (equatorial) Mass 318 ME Density 1.24
gm/cm3 Satellites gt 20 Structure Rocky Core
of 10-13 ME, surrounded by liquid metallic
hydrogen Temperature -148 C Magnetic Field
Huge Atmosphere 90 Hydrogen, 10 Helium
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From Brian Woodahl
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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
gt 20 Structure Similar to Jupiter Temperature
-178 C Magnetic Field Large Atmosphere 75
Hydrogen, 25 Helium
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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 gt 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
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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
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Neptune
Uranus
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Comparison of the Giant Planets
1.24
0.62
1.25
1.6
Mean density (gm/cm3)
http//www.freewebs.com/mdreyes3/chaptersix.htm
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Neptune
Jupiter
Uranus
Saturn
Log
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Pure H/He
Jupiter
CoRoT 9b
Saturn
50 H/He
10 H/He
Uranus
Pure Ice
Neptune
Pure Rock
Venus
Pure Iron
CoRoT 7b
Earth
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Reminder of what a transit curve looks like
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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) eiwt
Recall eiwt cos wt i sinwt
j1
X(t) is the time series
Power
Px(w) FTX(w)2
N0 number of points
2
1
2
(
(

)
)
S
S
Px(w)
Xj cos wtj
Xj sin wtj
N0
A DFT gives you as a function of frequency the
amplitude (power amplitude2) of each sine
function that is in the data
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Every function can be represented by a sum of
sine (cosine) functions. The FT gives you the
amplitude of these sine (cosine) functions.
A pure sine wave is a delta function in Fourier
space
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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.
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A Pictoral Catalog of Fourier Transforms
Time/Space Domain
Fourier/Frequency Domain
0
Time
Frequency (1/time)
Period 1/frequency
Comb of Shah function (sampling function)
x
1/x
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Time/Space Domain
Fourier/Frequency Domain
Positive frequencies
Negative frequencies
Cosine is an even function cos(x) cos(x)
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Time/Space Domain
Fourier/Frequency Domain
Sine is an odd function sin(x) sin(x)
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Time/Space Domain
Fourier/Frequency Domain
epx2
eps2
w
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)
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Time/Space Domain
Fourier/Frequency Domain
Note these are the diffraction patterns of a
slit, triangular and circular apertures
All functions are interchangeable. If it is a
sinc function in time, it is a slit function in
frequency space
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Fourier Transforms Convolution
Convolution
? f(u)f(xu)du f f
f(x)
f(x)
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Fourier Transforms Convolution
g(x)
Convolution is a smoothing function
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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 Fourier
Space fg
F ? G
f ? g
F G
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Alias periods
Undersampled periods appearing as another period
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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
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What does a transit light curve look like in
Fourier space?
In time domain
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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
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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
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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?
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In time space
convolution
X

In frequency space
X
Sinc of data window
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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.
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Frequencies repeat
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.
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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
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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.
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The effects of noise in your data
Signal level
Little noise
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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?
Frequency (c/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.
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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)
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A Flow Diagram for making exciting discoveries
1. Is there a periodic signal in my data?
no
Stop
yes
2. Is it due to Noise?
yes
Stop
no
3. What is its Nature?
4. Is this interesting?
no
Find another star
yes
5. Publish results
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Period Analysis with Lomb-Scargle Periodograms
LS Periodograms are useful for assessing the
statistical signficance of a signal
Px(w)

tan(2wt)
(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?)
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Fourier Transform
Scargle Periodogram
Amplitude (m/s)
Note Square this for a direct comparison to
Scargle power to power
FT and Scargle have different Power units
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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 lt n lt n2. This is given by
False alarm probability 1 (1eP)N NeP
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
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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).
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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 eP
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)
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Fourier Amplitude
Noisy data
In a normal Fourier transform the Amplitude of a
peak stays the same, but the noise level drops
Less Noisy data
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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.
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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.
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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 (1000-100000).
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)
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Least Squares Sine Fitting
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!
Fit a sine wave of the form y(t) Asin(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
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The first Tautenburg Planet HD 13189
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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
Amplitude (m/s)
Lomb-Scargle Periodogram Gives the power of each
frequency that is present in the data. Power is a
measure of statistical signficance
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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.
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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) n1 (2kL/np)

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Fitting a step functions with sines
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See file corot2b.dat for light curve
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See file corot7b.dat and corot7b.p04
PTransit 0.85 d 1.176 d