In this exercise you will set-up a simulation of the solar system in MATLAB and use Fourier Methods to analyse the output. If you are interested in making this part of your project please get in touch with me. For example you may be interested in researching the effects of tidal forces on the sun or the dynamics of binary star systems or the effects of black holes on solar systems.
The following files can be downloaded and put into your working directory in MATLAB:
Download the practical sheet here and work through it:
The above animation shows a computer simulation in MATLAB of the 8 planets and pluto orbiting around the sun.
In order to simulate the orbits of the planets, we make use of Newton's Law of Gravitation and treat each body as point masses. This enables us to calculate the force on each body in each dimension.
\[ \vec{F}_{i,j}=-G\frac{m_i m_j}{r_{i,j}^3}\vec{r_{i,j}} \]
Our coordinate system is cartesian based (i.e. \(x,y,z\)) so we must write down Newton's Law of gravitation using this system. The vector \(\vec{r_{i,j}}\) can be written down in column form as:
\[ \vec{r_{i,j}}=\left(\begin{array}{c} x_j-x_i \\y_j-y_i \\z_j-z_i \end{array} \right) \]
and thus its magnitude is
\[ |\vec{r_{i,j}}|=\sqrt{\left(x_j-x_i\right)^2 +\left(y_j-y_i\right)^2 +\left(z_j-z_i\right)^2} \]
Hence, the components of the force in the \(x, y\) and \(z\) direction are:\[ \left(\begin{array}{c} Fx_{i,j}\\ Fy_{i,j}\\ Fz_{i,j} \end{array}\right) =-G\frac{m_i m_j}{\left({\left(x_j-x_i\right)^2 +\left(y_j-y_i\right)^2 +\left(z_j-z_i\right)^2}\right)^{3/2}} \left(\begin{array}{c} x_j-x_i \\y_j-y_i \\z_j-z_i \end{array} \right) \]
For the motion of the planets we apply Newton's 2nd Law of motion, which states that the force applied to a mass is equal to its acceleration multiplied by its mass. Hence, Newton's Second Law of Motion enables us to use the force calculated from Newton's Law of Gravitation to the motion that the bodies take in space.
Newton's 2nd Law states for net force, \(\vec{F}\), applied to object mass, \(m\) causing the object to accelerate with acceleration, \(a\)
\[ \vec{F}=m\vec{a} \]
Hence, to solve for the motion of each object (i.e. find the \(x_i,y_i,z_i\) positions with time) we must solve three differential equations for each dimension:
\[ \begin{eqnarray} Fx_{i,j} &=& m_i\frac{d^2 x_i}{dt^2}\\ Fy_{i,j} &=& m_i\frac{d^2 y_i}{dt^2}\\ Fz_{i,j} &=& m_i\frac{d^2 z_i}{dt^2} \end{eqnarray} \]
MATLAB can solve differential equations using a variety of methods, but the one we choose for this exercise is ODE113
. However, this method can only solve 1st order differential equations (those having a first order derivative). Clearly the equations to solve are 2nd order differential equation, but we can turn them into 1st order differential equations by specifying another 3 equations for the 3 components of velocity, \(vx_{i}, vy_{i}, vz_{i}\), which give us the following 6 equations to solve for each body in the solar system:
\[ \begin{eqnarray} Fx_{i,j} &=& m_i\frac{d vx_i}{dt}\\ Fy_{i,j} &=& m_i\frac{d vy_i}{dt}\\ Fz_{i,j} &=& m_i\frac{d vz_i}{dt}\\ vx_i &=& \frac{d x_i}{dt}\\ vy_i &=& \frac{d y_i}{dt}\\ vz_i &=& \frac{d z_i}{dt} \end{eqnarray} \]
Take a look at the solve_solar_system.m
file, taking care to read the comments and follow roughly what the code does.
You run the code by making sure the solve_solar_system.m
file is in your working directory typing:
[t,y,bodies]=solve_solar_system;
The output is as follows:
t
is the time in seconds for the simulation bodies
is a cell of strings that contains the names of each of the bodies simulated. bodies
cell. y
matrix are the x, y and z positions and the ux, uy and uz velocities of the sun. Then it repeats, mercury, venus and so on.
dist=y(:,[1:3]+3*6)-y(:,[1:3]); % this is the distance between x, y and z
dist_mod=sqrt(sum(dist.^2,2)); % this is the magnitude of the vector (sqrt of sum of squares of x, y and z)
plot(t,dist_mod);
There is another m-file called animate_solar_system.m
. After running solve_solar_system.m
you can then run this m-file to see an animation of the solar system.
Type:
animate_solar_system
If you wish to break out of the animation at any time, go to the MATLAB command prompt and type Ctrl-X
.
The data generated lend naturally to Fourier analysis because they are periodic. We will now cover some fundamentals of Fourier analysis
The Fourier series is a powerful concept in the analysis of waves. Fourier methods are used in many different situations, such as in seismology, many topics in physics, spectroscopy and acoustics.
Harmonic waves are those that are described by trigonometric functions: sines and cosines. We know how to deal with these functions mathematically, but other functions such as square waves, or saw-tooth waves (so called aharmonic functions) are more problematic. However, it turns out that we can build up any periodic function by adding lots of sines and cosine functions together.
Lets say for example we have a periodic wave that looks like a saw-tooth pattern. The animation below shows that starting with a sine wave of the same period and adding sine waves of higher frequencies (and lower amplitudes) we may eventually build up something similar to a saw-tooth wave.
Another example is given here, this time a saw-tooth wave is represented by the first 5 harmonic functions (source: Wikipedia).
Similarly see the representation of a square wave below. We can build up a square wave by adding sine functions of different amplitudes together.
The above animation shows the approximation of a square wave by the sum of 6 harmonic functions (source: Wikipedia).
The basis of how Fourier methods work are the following results
\begin{eqnarray} \int _{-\frac{\tau}{2}}^\frac{\tau}{2} \sin\left(\frac{2\pi mt}{\tau}\right)\sin\left(\frac{2\pi nt}{\tau}\right) dt&=&\frac{\tau}{2}\delta _{m,n}\\ \int _{-\frac{\tau}{2}}^\frac{\tau}{2} \cos\left(\frac{2\pi mt}{\tau}\right)\cos\left(\frac{2\pi nt}{\tau}\right) dt&=&\frac{\tau}{2}\delta _{m,n}\\ \int _{-\frac{\tau}{2}}^\frac{\tau}{2} \sin\left(\frac{2\pi mt}{\tau}\right)\cos\left(\frac{2\pi nt}{\tau}\right) dt&=&0\\ \int _{-\frac{\tau}{2}}^\frac{\tau}{2} \sin\left(\frac{2\pi mt}{\tau}\right) dt&=&0\\ \int _{-\frac{\tau}{2}}^\frac{\tau}{2} \cos\left(\frac{2\pi mt}{\tau}\right) dt&=&0 \end{eqnarray}
Where \(\tau\) is the period of the wave being represented, \(\delta _{m,n}\) is equal to unity when \(m=n\) and zero otherwise. The above integrals state that the product of two waves, when integrated over one full period, is equal to zero unless they have the same frequency and phase. They also say that the integral of a harmonic wave over a full period is equal to zero. It is this property that enables Fourier methods to work.
It turns out that any periodic function can be described using an infinite sum of harmonic (i.e. sines and cosines) function as follows:
\[ f\left(t\right)=\frac{a_0}{2}+\sum a_n\cos\left(\frac{2\pi nt}{\tau} \right)+\sum b_n\sin\left(\frac{2\pi nt}{\tau}\right) \]
This is known as a Fourier series. The coefficients, \(a_0,a_n\) and \(b_n\), are known as Fourier coefficients and are dependent on the function being approximated, \(f\left(t\right)\). However, if we now make use of the mathematical results above we may determine formulae for these coefficients.
Firstly, we integrate both sides of the Fourier series over the interval \(-\frac{\tau}{2}\) to \(\frac{\tau}{2}\)
\[ \begin{eqnarray} \int _{-\frac{\tau}{2}}^\frac{\tau}{2} f\left(t\right) dt&=&\int _{-\frac{\tau}{2}}^\frac{\tau}{2} \frac{a_0}{2} dt+\int _{-\frac{\tau}{2}}^\frac{\tau}{2} \sum a_n\cos\left(\frac{2\pi nt}{\tau} \right)+\sum b_n\sin\left(\frac{2\pi nt}{\tau}\right) dt\\ \int _{-\frac{\tau}{2}}^\frac{\tau}{2} f\left(t\right) dt&=& a_0\times \frac{\tau}{2} \end{eqnarray} \]
where the sums within the integral on the first line are zero through the mathematical results above. We arrive at the result:
\[ a_0=\frac{2}{\tau}\int _{-\frac{\tau}{2}}^\frac{\tau}{2} f\left(t\right) dt \]
Similarly we can derive an expression for the \(a_n\) coefficient by multiplying the Fourier series by \(cos\left(nt\right)\) and integrating both sides over one period:
\[ \begin{eqnarray} \int _{-\frac{\tau}{2}}^\frac{\tau}{2} f\left(t\right)\cos\left(\frac{2\pi nt}{\tau}\right) dt&=&\int _{-\frac{\tau}{2}}^\frac{\tau}{2} \frac{a_0}{2}\cos\left(\frac{2\pi nt}{\tau}\right) dt+\int _{-\frac{\tau}{2}}^\frac{\tau}{2} \sum a_n\cos\left(\frac{2\pi nt}{\tau} \right)\cos\left(\frac{2\pi nt}{\tau}\right)+\sum b_n\sin\left(\frac{2\pi nt}{\tau}\right)\cos\left(\frac{2\pi nt}{\tau}\right) dt\\ \int _{-\frac{\tau}{2}}^\frac{\tau}{2} f\left(t\right)\cos\left(\frac{2\pi nt}{\tau}\right) dt&=&a_n\times \frac{\tau}{2} \end{eqnarray} \]
This enables us to define the \(a_n\) coefficients as:
\[ a_n = \frac{2}{\tau} \int _{-\frac{\tau}{2}}^\frac{\tau}{2} f\left(t\right)\cos\left(\frac{2\pi nt}{\tau}\right) dt \]
Multiplying the Fourier series by \(sin\left(\frac{2\pi nt}{\tau}\right)\) and integrating over one period yields the \(b_n\) coefficients:
\[ b_n = \frac{2}{\tau} \int _{-\frac{\tau}{2}}^\frac{\tau}{2} f\left(t\right)\sin\left(\frac{2\pi nt}{\tau}\right) dt \]
Therefore, we can describe any periodic function as an infinite sum of harmonic functions with the amplitudes given by the integrals above. This is useful as it enables us to define the frequencies that are present in an oscillation.
The \(a_n\) and \(b_n\) coefficients in the Fourier series tell us the frequencies of harmonic oscillations that are present in a time-series; however, the frequencies are integer multiples of the frequency of the original wave. In general we need a representation that can have a continuum of frequency values. This is called the Fourier Transform.
MATLAB has functions fft
and ifft
that enable us to do these Fourier Transforms.
Lenard Euler derived a formula relating sines, cosines, the exponential constant and an imaginary number equal to the square-root of -1, \(i\)
\[ \exp\left(i x\right)=\cos\left( x\right)+i\sin\left(x \right) \]
Hence, we can generalise the Fourier series using the following integral:\[ \mathcal{F}\left(\nu \right)=\int _{-\infty}^\infty f\left(t\right)\exp\left(-i2\pi\nu t \right) dt \]
This is known as the Fourier Transform. \(\mathcal{F}\left(\nu \right)\) is a complex number whose real part contains the amplitudes of the cosine harmonic functions representing the function \(f\left(t\right)\) and the imaginary part contains the amplitudes of the sinusoidal harmonic functions. The frequency, \(\nu\), can take on any value in the Fourier transform, rather than be integer multiples of the frequency of the initial wave.Since \(\mathcal{F}\left(\nu\right)\) contains the amplitudes of the harmonic oscillations we may also define the inverse Fourier transform, which gets us back to the original function:
\[ f\left(t\right)=\int _{-\infty}^\infty \mathcal{F}\left(\nu \right)\exp\left(i2\pi\nu t \right) d\nu \]
fft
- Fast Fourier Transform ifft
- inverse Fast Fourier Transform
Let us experiment with Fourier transforms in MATLAB. First of all try the following code segment, which will generate some times and a harmonic oscillation with frequency 1 Hz:
tsamp=0.0001;
tf=0:tsamp:100;
nu=1;
f=10.*sin(2.*pi.*tf.*nu);
figure('name','Time-series of harmonic wave');
plot(tf,f)
Next, perform the Fourier transform and plot it out:
F=fft(f);
L=length(f);
freq=1./tsamp.*linspace(0,1,L);
figure('name','FFT of harmonic wave');
plot(freq,2.*abs(F)./L);
set(gca,'xscale','log');
Note, that you will find two peaks at an amplitude of 10. The one at 1 Hz frequency is the true frequency and the one at a frequency of 10^{4} is due to aliasing. It is an artifact and should be ignored because the fft
will not work for frequencies equal to about the sampling frequency (i.e. 1./tsamp
).
Next we plot out the original function, then do the inverse Fourier transform to show they are equal.
finv=ifft(F);
You should see that both curves are equal to each other.
figure('name','Original harmonic wave and ifft');
plot(tf,f);hold on;
plot(tf,finv,'r');
Next try the same, but adding two harmonic waves together, one with a frequency of 1 Hz and amplitude of 10, and the other with frequency of 10 Hz and amplitude of 5.
tsamp=0.0001;
tf=0:tsamp:100;
nu1=1;
nu2=10;
f=10.*sin(2.*pi.*tf.*nu1)+5.*sin(2.*pi.*tf.*nu2);
figure('name','Time-series of two harmonic waves');
plot(tf,f)
Next, perform the Fourier transform and plot it out:
F=fft(f);
L=length(f);
freq=1./tsamp.*linspace(0,1,L);
figure('name','FFT of sum of two harmonic waves');
plot(freq,2.*abs(F)./L);
set(gca,'xscale','log');
You should see that the resulting Fourier transform has a peak at 1 Hz of amplitude 10 and a peak at 10 Hz at an amplitude of 5.
If you would like to initialize the model with different data the positions and velocities (Ephemeris data) of the planets at different times can be retrieved by email from the JPL Horizons system . Click on the link to send an email, you will need to do it for each planet in turn.
You should see that the email has the subject Job
and that the message body is
!$$SOF
EMAIL_ADDR=''
START_TIME = '2014-Sep-30 00:00:00'
STOP_TIME = '2014-Sep-30 00:00:01'
TABLE_TYPE = 'Vector'
REF_PLANE = 'Ecliptic'
CENTER = '@010'
COMMAND='999'
!$$EOF
This means get the position and velocity on the 2014-Sep-30 at 12 o'clock midnight in a coordinate system where the sun is centred on (0,0,0). i.e. the centre of the coordinates is the object 010, which refers to the sun. The value of COMMAND
is set to 999 which means get the data for Pluto. Mercury is 199, Venus is 299 and so on...
When you send the email, click on the tab below to see the reply you will get from the JPL Horizons system.
Automated mail xmit by MAIL_REQUEST, PID= 14032 Mon Sep 29 16:07:43 2014
++++++++++++++++++++++++++++++++ (part 1 of 1) +++++++++++++++++++++++++++++++
*******************************************************************************
Revised: Dec 30, 2013 134340 Pluto 999
Fit to all available observations (PLU043/DE431)
ICRF/J2000.0 helio. ecliptic osc. elements (AU, DAYS, DEG, period=Julian yrs):
EPOCH= 2456656.5 ! 2013-Dec-30.00 (CT)
EC= 2.479750960904014E-01 QR= 2.950229731448114E+01 IN= 1.733434183305855E+01
OM= 1.101908708982876E+02 W = 1.121892886119843E+02 Tp= 2447571.800166774075
A = 3.923047915182824E+01 AD= 4.895866098917534E+01 PR= 8.974987331741650E+04
PHYSICAL DATA (updated 2013-Nov-19, Mc= Charon mass, radius is IAU 2009):
Mass Pluto (10^22 kg) = 1.307+-0.018 Density Pluto:
GM (planet) km^3/s^2 = 872.4 (R = 1195 km) = 1.83 g/cm^3
Mass ratio (Mc/Mp) = 0.117 Radius of Pluto, Rp = 1195 km
Surface gravity = 61.1 cm/s^2 Geometric albedo = 0.3
Sidereal orbit period = 249.58932 yr Mean orbit velocity = 4.656 km/s
Escape velocity (km/s)= 1.21
*******************************************************************************
*******************************************************************************
Ephemeris / MAIL_REQUEST Mon Sep 29 16:07:43 2014 Pasadena, USA / Horizons
*******************************************************************************
Target body name: Pluto (999) {source: PLU043}
Center body name: Sun (10) {source: DE-0431LE-0431}
Center-site name: BODY CENTER
*******************************************************************************
Start time : A.D. 2014-Sep-30 00:00:00.0000 CT
Stop time : A.D. 2014-Sep-30 00:00:01.0000 CT
Step-size : 60 minutes
*******************************************************************************
Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}
Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}
Center radii : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}
Output units : KM-S
Output format : 03
Reference frame : ICRF/J2000.0
Output type : GEOMETRIC cartesian states
Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch
*******************************************************************************
JDCT
X Y Z
VX VY VZ
LT RG RR
*******************************************************************************
$$SOE
2456930.500000000 = A.D. 2014-Sep-30 00:00:00.0000 (CT)
1.063301456466383E+09 -4.775737235886205E+09 2.033982415147326E+08
5.404835952771680E+00 5.598716109091834E-02 -1.594326579818693E+00
1.633430609314200E+04 4.896901773387416E+09 1.052769074493617E+00
$$EOE
*******************************************************************************
Coordinate system description:
Ecliptic and Mean Equinox of Reference Epoch
Reference epoch: J2000.0
xy-plane: plane of the Earth's orbit at the reference epoch
x-axis : out along ascending node of instantaneous plane of the Earth's
orbit and the Earth's mean equator at the reference epoch
z-axis : perpendicular to the xy-plane in the directional (+ or -) sense
of Earth's north pole at the reference epoch.
Symbol meaning
JDCT Epoch Julian Date, Coordinate Time
X x-component of position vector (km)
Y y-component of position vector (km)
Z z-component of position vector (km)
VX x-component of velocity vector (km/sec)
VY y-component of velocity vector (km/sec)
VZ z-component of velocity vector (km/sec)
LT One-way down-leg Newtonian light-time (sec)
RG Range; distance from coordinate center (km)
RR Range-rate; radial velocity wrt coord. center (km/sec)
Geometric states/elements have no aberration corrections applied.
Computations by ...
Solar System Dynamics Group, Horizons On-Line Ephemeris System
4800 Oak Grove Drive, Jet Propulsion Laboratory
Pasadena, CA 91109 USA
Information: http://ssd.jpl.nasa.gov/
Connect : telnet://ssd.jpl.nasa.gov:6775 (via browser)
telnet ssd.jpl.nasa.gov 6775 (via command-line)
Author : Jon.Giorgini@jpl.nasa.gov
*******************************************************************************
Note that the MATLAB file solve_solar_system.m
above already has the 3d position and velocity data for all the planets. You may want to use initialisation data for a different date, in which case generate the Ephemeris data for all planets (as described above) and put the numbers \(\left(X,Y,Z, VX,VY,VZ\right)\), in the solve_solar_system.m
file (lines 101 to 119). The first element of each array is the Sun, the second Mercury, then Venus and so on.