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

Crust:

1. Silicate Mantle

Nickel-Iron Core

Venus is believed to have an internal structure similar to the Earth

Internal Structure of Venus

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

The moon has a very small core, but a large mantle (≈70%)

Internal Structure of the Moon

Comparison of Terrestrial Planets

http://astronomy.nju.edu.cn/~lixd/GA/AT4/AT411/HTML/AT41105.htm

R = 0.28 REarth

M = 0.015 MEarth

= 3.55 gm cm–3

R = 0.25 REarth

M = 0.083MEarth

= 3.01 gm cm–3

R = 0.41 REarth

M = 0.025MEarth

= 1.94 gm cm–3

R = 0.38 REarth

M = 0.018 MEarth

= 1.86 gm cm–3

Note: The mean density increases with increasing distance from Jupiter

Satellites

Internal Structure of Titan

Mercury

MarsVenus

Earth

Moon

1

2

3

4

5

7

10

0.2 0.4

Radius (REarth)

(g

m/c

m3)

0.6 0.8 1 1.2 1.4 1.6 1.8 2

No iron

Earth-likeIron enriched

From Diana Valencia

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

NeptuneUranus

http://www.freewebs.com/mdreyes3/chaptersix.htm

Comparison of the Giant Planets

1.24 0.62 1.25 1.6

Mean density (gm/cm3)

Jupiter

Saturn

Neptune

Uranus

Log

CoRoT 7b

CoRoT 9b

Jupiter

Saturn

Uranus

Neptune

Earth

Venus

Pure H/He

50% H/He

10% H/He

Pure Ice

Pure Rock

Pure Iron

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)

FT() = Xj (t) e–it

N0

j=1

A DFT gives you as a function of frequency the amplitude (power = amplitude2) of each sine function that is in the data

Power: Px() = | FTX()|2

1

N0

Px() =

1

N0

N0 = number of points

[( Xj cos tj + Xj sin tj ) ( ) ]2 2

Recall eit = cos t + i sint

X(t) is the time series

A pure sine wave is a delta function in Fourier space

t

P

Ao

FT

Ao

1/P

Every function can be represented by a sum of sine (cosine) functions. The FT gives you the amplitude of these sine (cosine) functions.

Fourier Transforms

Two important features of Fourier transforms:

1) The “spatial or time coordinate” x maps into a “frequency” coordinate 1/x (= s or )

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 Fourier/Frequency Domain

Comb of Shah function (sampling function)

x 1/x

Time Frequency (1/time)

Period = 1/frequency

0

Time/Space Domain Fourier/Frequency Domain

Cosine is an even function: cos(–x) = cos(x)

Positive frequencies

Negative frequencies

Time/Space Domain Fourier/Frequency Domain

Sine is an odd function: sin(–x) = –sin(x)

Time/Space Domain Fourier/Frequency Domain

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 (-function) the Fourier transform is infinitely wide (constant)

w 1/w

e–x2 e–s2

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

Convolution

Fourier Transforms : Convolution

f(u)(x–u)du = f *

f(x):

(x):

Fourier Transforms: Convolution

(x-u)

a1

a2

g(x)a3

a2

a3

a1

Convolution is a smoothing function

2) In Fourier space the convolution is just the product of the two transforms:

Normal Space Fourier Space f*g F G

Fourier Transforms

The second important features of Fourier transforms:

f g F * G

sinc 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 t, the shortest frequency you can detect with no aliases is 1/(2t)

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 + 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)?

This is a sync function caused by the length of the data window

P = 3.85 d= 0.26 c/d

A longer time string of the same sine

A short time string of a sine

Wide sinc function

Narrow sinc function

Sine times step function of length of your data window

-fnc * step

The peak of the combs is modulated with a shape of another sinc function. Why?

What happens when you carry out the Fourier transform of our Transit light curve to higher frequencies?

= * XTransit shape Comb

spacing of P

Length of data string

In time „space“

In frequency „space“

X

* = convolution

=

Sinc of data window

Sinc function of transit shape

Comb spacing of 1/P

*

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.

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.

Nyquist

Frequencies repeat

This pattern gets repeated in intervals of 200 c/d for this sampling. Frequencies on either side of the peak are – and +

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

More noise

A lot of noise

Noise level

Signal level

Frequency (c/d)

Transit period of 3.85 d (frequency = 0.26 c/d)

20 d?

Time (d)

20 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.

This is the previous transit light curve with more realistic sampling typical of what you can achieve from the ground.

The Effects of Sampling

A very nice sine fit to data….

That was generated with pure random noise and no signal

P = 3.16 d

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?

2. Is it due to Noise?

3. What is its Nature?

yesStop

no

4. Is this interesting?

noStopyes

yesFind another starno

5. Publish results

A Flow Diagram for making exciting discoveries

Period Analysis with Lomb-Scargle Periodograms

LS Periodograms are useful for assessing the statistical signficance of a signal

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?)

1

2Px() =

[ Xj sin tj–]2

j

Xj sin2 tj–

[ Xj cos tj–]2

j

Xj cos2 tj–j

+1

2

tan(2) = sin 2tj)/cos 2tj)j j

Fourier Transform Scargle Periodogram

Am

plit

ude

(m/s

)

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

False alarm probability ≈ 1 – (1–e–P)N ≈ Ne–P

N = number of indepedent frequencies ≈ number of data points

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 1 < < 2. This is given by:

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)

In a normal Fourier transform the Amplitude of a peak stays the same, but the noise level drops

Noisy data

Less Noisy data

Fourier 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.

versus Lomb-Scargle Amplitude

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, , 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 2 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 2 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 2 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 2 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(t + ) + ConstantWhere = 2/P, = phase shiftBest fit minimizes the 2:

2 = di –gi)2/N

di = data, gi = fit

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/

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!

The first Tautenburg Planet: HD 13189

Least squares sine fitting: The best fit period (frequency) has the lowest 2

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

Am

plit

ude

(m/s

)

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/n)

Fitting a step functions with sines

See file corot2b.dat for light curve

Prot = 23 d

See file corot7b.dat and corot7b.p04

0.035%

PTransit = 0.85 d = 1.176 d