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 10 13 cm) Period: 1 year Radius: 1 R E (6378 km) Mass: 1 M E (5.97 10 27 gm) Density 5.50 gm/cm 3 (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, CO 2, water
Internal Structure of the Earth
Venus Distance: 0.72 AU Period: 0.61 years Radius: 0.94 R E Mass: 0.82 M E Density 5.4 gm/cm 3 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 R E Mass: 0.11 M E Density: 4.0 gm/cm 3 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% CO 2, 3% Nitrogen, argon, traces of oxygen
Internal Structure of Mars
Mercury Distance: 0.38 AU Period: 0.23 years Radius: 0.38 R E Mass: 0.055 M E Density 5.43 gm/cm 3 (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 R E Mass: 0.011 M E Density: 3.34 gm/cm 3 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 R Earth M = 0.015 M Earth ρ = 3.55 gm cm 3 R = 0.25 R Earth M = 0.083M Earth ρ = 3.01 gm cm 3 Note: The mean density increases with increasing distance from Jupiter R = 0.41 R Earth M = 0.025M Earth ρ = 1.94 gm cm 3 R = 0.38 R Earth M = 0.018 M Earth ρ = 1.86 gm cm 3 http://astronomy.nju.edu.cn/~lixd/ga/at4/at411/html/at41105.htm
Internal Structure of Titan
ρ (gm/cm 3 ) 10 7 5 4 3 Mercury Moon Mars Iron enriched Venus Earth No iron Earth-like 2 From Diana Valencia 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Radius (R Earth )
Jupiter Distance: 5.2 AU Period: 11.9 years Diameter: 11.2 R E (equatorial) Mass: 318 M E Density 1.24 gm/cm 3 Satellites: > 20 Structure: Rocky Core of 10-13 M E, 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 R E (equatorial) = 0.84 R J Mass: 95 M E (0.3 M J ) Density 0.62 gm/cm 3 (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 R E (equatorial) = 0.36 R J Mass: 14.5 M E (0.05 M J ) Density: 1.25 gm/cm 3 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 R E (equatorial) = 0.35 R J Mass: 17 M E (0.05 M J ) Density: 1.6 gm/cm 3 (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 http://www.freewebs.com/mdreyes3/chaptersix.htm 1.24 0.62 1.25 1.6 Mean density (gm/cm 3 )
Neptune Uranus Jupiter Saturn Log
Pure H/He Saturn Jupiter CoRoT 9b H/He dominated planets Uranus Neptune Venus CoRoT 7b Earth 10% H/He Pure Iron Pure Rock 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) N 0 FT(ω) = X j (t) e iωt j=1 1 Power: P x (ω) = FT X (ω) 2 P x (ω) = 1 N 0 N 0 Recall e iωt = cos ωt + i sinωt X(t) is the time series N 0 = number of points 2 2 [( Σ X j cos ωt) j + ( Σ X j sin ωt )] j A DFT gives you as a function of frequency the amplitude (power = amplitude 2 ) 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. P FT A o A o t 1/P ω A pure sine wave is a delta function in Fourier space
Fourier Transforms Two important features of Fourier transforms: 1) The spatial or time coordinate x maps into a frequency coordinate 1/x (= s or ν) 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 Time 0 Frequency (1/time) Period = 1/frequency Comb of Shah function (sampling function) x 1/x
Time/Space Domain Fourier/Frequency Domain Negative frequencies Positive frequencies Cosine is an even function: cos( x) = cos(x)
Time/Space Domain Fourier/Frequency Domain Sine is an odd function: sin( x) = sin(x)
Time/Space Domain Fourier/Frequency Domain e πx 2 e πs 2 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 (δ-function) the Fourier transform is infinitely wide (constant)
Time/Space Domain Fourier/Frequency Domain All functions are interchangeable. If it is a sinc function in time, it is a slit function in frequency space Note: these are the diffraction patterns of a slit, triangular and circular apertures
Fourier Transforms : Convolution Convolution f(u)φ(x u)du = f * φ f(x): φ(x):
a 3 a 2 Fourier Transforms: Convolution φ(x-u) a 1 a 2 a 3 g(x) a 1 Convolution is a smoothing function
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 Fourier Space F G F * G sinc sinc 2
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 Δt, the shortest frequency you can detect with no aliases is 1/(2Δt) 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 ν + 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)? P = 3.85 d ν = 0.26 c/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 δ-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 = Sinc function of transit shape X 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 ν and +ν 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) 20 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 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? yes no Stop 2. Is it due to Noise? A Flow Diagram for making exciting discoveries yes no Stop 3. What is its Nature? 4. Is this interesting? yes no 5. Publish results Find another star
Period Analysis with Lomb-Scargle Periodograms LS Periodograms are useful for assessing the statistical signficance of a signal P x (ω) = 1 2 [ Σ X j cos ω(t j τ) ] 2 j Σ X j cos2 ω(t j τ) j + 1 2 [ Σ X j sin ω(t j τ) ] 2 j Σ X j sin2 ω(t j τ) tan(2ωτ) = (Σsin 2ωt j )/ (Σcos 2ωt j ) 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?)
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
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 ν 1 < ν < ν 2. 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, N i, and the number of data points, N 0 : N i = 6.362 + 1.193 N 0 + 0.00098 N 0 2
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 In a normal Fourier transform the Amplitude of a peak stays the same, but the noise level drops Less Noisy data
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, σ, 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 χ 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 χ 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 χ 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 χ 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(ωt + φ) + Constant Where ω = 2π/P, φ = phase shift Best fit minimizes the χ 2 : χ 2 = Σ (d i g i ) 2 /N d i = data, g i = 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
Least squares sine fitting: The best fit period (frequency) has the lowest χ 2 Amplitude (m/s) 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 B n = ( 1) n+1 (2kL/nπ)
Fitting a step functions with sines
See file corot2b.dat for light curve
See file corot7b.dat and corot7b.p04 P rot = 23 d 0.035% P Transit = 0.85 d = 1.176 d