## The frequency analysis:

Through the following link, you can acces a C-language code which implements the Modified Fourier Transform (Laskar 1999) and its variants with frequency corrections: fmft.cYou will need to compile it with nrutil.c and the header nrutil.h , which contain the routines of C-language dynamical declarations. After downloading the files the compilation is typically:

gcc -c nrutil.cgcc -c fmft.cThen, the resulting

*.ofiles must be linked with a driver file:

gcc -o main_fmft main_fmft.c fmft.o nrutil.o -lmAn example of the driver file: main_fmft.c . This particular driver uses

fmft.cto determine frequencies, amplitudes and phases in a time series resulting from the secular evolution of planetary orbits. Typically, you type:

main_fmft < input > outputwhere

inputis the file containing the time series andoutputwill contain results of the frequency analysis. The fileinputmust have 3 columns(time x y)and more thanNDATArows (see below). The complex time seriesz(time)= x(time) + i y(time)must have an equal time spacing. Inmain_fmft.c, one shell define:The

- interval of frequencies (in arcsec per year) where the frequencies are calculated
(MIN_F,MAX_F)- variant of the frequency analysis (an integer
FMFT_FLAG, seefmft.cfor options)- number of frequencies to be calculated
(NFREQ)- spacing in the input time series
(DATA_SEP)- length of the time series
(NDATA), must be a power of 2 and the number of rows in theinputfile must be equal or larger thanNDATAoutputfile will have 3 columns (frequency, amplitude and phase) andNFREQrows.An example of input and output files. This example is

e*exp(i*varpi), whereeandvarpiare the eccentricity and perihelion longitude of Jupiter sampled each 1000 yr for 10 Myr(NDATA <= 10000). The time series were artificially limited to 10 harmonics with largest amplitudes.

If nothing works, contact me on:

david@obs-nice.fr

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