Download final paper written on this project here

Cris Cecka
April 4, 2005
Computational Physics
Professor Saeta
Fourier Transforms of Semistable Orbits
Introduction
Previously, we developed an integrator to model a planet of
negligible mass being acted on by gravity from two stars in a binary
system comprising Alpha-Centauri. This was accomplished by
integrating the angular equation for the binary system independent of
the planet’s position and integrating the equations of motion for the
planet with respect to the gravitational potential of the two stars.
Previously, we investigated the orbits of such a planet and found two
particularly stable classes of orbits: those which are tightly bound to
a single star and those which are weakly bound to the combined mass
of the two stars. In this problem, we will investigate the stability of
these classes of orbits using the Fourier transform.
Background
In the case of the tightly bound orbit which is nearly circular about one of the stars, we can make some
assumptions about the relative path of the planet. Its relative coordinates
( x, y )  ( X p , Y p )  ( X s , Y s )
will have the form
y (t )  r sin( t   )
x(t )  r cos(t   )
where r is the radius of the orbit, and ω is the angular frequency which is given by
r 2 
Fp
GM
v2
 ap 
 2s
r
Mp
r
Thus, a Fourier transform in time of the x(t) or y(t) position of this orbit would yield a power spectrum with a single
peak at the frequency
f 
1
2
GM s
r3
and if we were to use units of A.U., years, and M☼ (Sun Masses [unitless]), we would have
f [1 / yr ] 
M s A.U .3 1

T
r 3 yr 2
where T is the period of this orbit. However, due to the motion of the star itself and the force from the opposite star
in the system, the orbit cannot be perfectly circular. Consider the case of a closed non-circular orbit. We know that
each term in the relative position of the planet (x(t), y(t)) all must have the same periodicity with fundamental
frequency f  1 T . Thus, in this case, the Fourier transform can only yield peaks at f and the harmonics of f.
Finally, the Fourier transform will give us the most information in the case of general non-closed orbits in regards to
its stability and structure since we have now determined some simple structures to look for. The power spectrum of
a non-closed orbit will tell us the ‘effective period’ of the orbit as well as an image of the stability. The sharper the
peaks, the more predictable and stable the orbit of the planet. On the other hand, ‘smeared peaks’ and randomness in
the spectrum tells us that the path does not have string periodicity, is more chaotic, and more likely to be involved in
a collision or divergent trajectory.
Strategy
In order to get an accurate Fourier transform of this data, we must ensure that the assumptions of the
Fourier transform are met. First, the sampling interval must be constant. The RKF algorithm that we are using to
integrate the equations of motion uses an adaptive step size to provide that accuracy requested by the user. Smaller
steps sizes are used for more error-prone integration sections (such as small orbits and tight turns) while larger steps
are used for less error-prone sections (such as with 0 potential, free particle states). This must be undone in order to
perform the Fourier transform. Preserving the adaptive step size in the interest of accuracy and speed, we must
interpolate the results to achieve constant steps in the data. If t is the array of times with constant step size which we
would like to have, and xp, xt are the arrays with non-constant step size dt that we are given by the output of the
RKF algorithm, the function call x = interp1(tp, xp, t, 'spline') will interpolate the function
f (tp)  xp using a ‘natural’ cubic spline algorithm of the underlying function f (T )  X to create the discrete
function f (t )  x . An alternative to calling interp1 would be using the function call
sol = ode45(odefun,[t0 tf],y0,options);
and using the function x = deval(sol, t) to accomplish the interpolation. Since these methods are very
nearly equivalent, we will use the first because of its larger portability.
Second, the function must be sampled at a frequency high enough for the Nyquist frequency f N , the
highest frequency the Fourier transform can detect, to be equal to or greater than twice the highest waveform
frequency  , f N  2 . Without this condition, the spectrum can be aliased or flipped. In order to meet it with
high probability, we’ll use the same number of points in the transform as were used in the integration. Thus, unless
there is an extremely large and sharp spike where nearly ALL of the integration steps were taken and upon our
interpolation this spike is ‘missed’ or skipped over, this condition should be met easily. If at any point we feel that
this is not being met, a simple fix would be to increase the tolerance levels on the integrator to force smaller step
sizes.
Finally, the Fourier transform assumes periodicity of the signal. Thus, in the case that y (beg )  y (end )
and y ' (beg )  y ' (end ) , there are discontinuities in the signal which will cause high frequencies to be required in
attempting to match these discontinuities. One solution to this is windowing, a technique in which the signal is
amplitude modulated such that the end points and derivatives are forced to 0. Although this can distort the output
slightly since we are changing the signal, the gain in the sharpness and accuracy of the power spectrum is greater as
shown below.
Note that the broadened peak of the original signal hides the presence of the harmonic and that only after modulating
it to reduce this broadening does it become clearly apparent (though it does exist in the original as well). It should be
mentioned that the frequencies and relative magnitudes of all peaks is naturally preserved in the modulation and that
its only effect is to decrease broadening.
Tight Orbits about a Single Star
Here, we will investigate tight orbits in which the relative position of the planet can trace a very circular
path when r is small since the effect of the opposing star becomes negligible. An orbit with approximate radius of
2A.U. is shown in the introduction to this paper. It can be created with the initial conditions
( X p , Y p )  (6.8532,0) , (V X , VY )  (0,3.2466) . From this, we can extract the relative position of the
p
p
planet with respect to the star it orbits and perform the Fourier Transform on it:
Here, the peak occurs at the frequency
.3750  f 
1.10
 .3708 . A smaller peak occurs at the frequency
23
.7500  2 f , the first harmonic! This tells us that the orbit is not entirely circular, but actually has a slight average
oblate behavior to it. The relative power in these two harmonics is p1 / p 2  19874.67 / .0900  220978 in the
x-direction, and a relative power of 232771 in the y-direction. This should be impossible to distinguish and the
orbit can be considered circular. Even with perfect reinforcement on each axis, the eccentricity could not exceed
a 2  b2


a
2  2
 
2
220978  2  2
2  2
220978

232771

2
 .091
However, this could change over time and is a good first hint of instability. Looking at the same system after 1600
years, we see that this oblateness has been magnified (what part of this is natural and what part is numerical
inaccuracy is a non-trivial question) over time.
Here, after 1600 years, the harmonics are now distinctly visible, and the eccentricity can be seen in the Relative
Position itself.
Two interesting questions then arise. How does this instability develop over time and how does the stability depend
on how tightly the planet is bound to the star. Logically, there must be a point at which the influence of the opposing
star ceases to be negligible in both time and space. Here, we will investigate the dependence over space. The
following graph shows the Power Spectra of various bound states of the planet to the more massive star.
Orbit R
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
Each increment in orbital radius is offset for easy viewing as well as amplified by a power of 10 each time since the
2/3
absolute power of the fundamentals drops off each time. We can clearly see the f ~ R
dependence here in the
drift of the fundamental frequency. Primarily, we’re interested in the relative power of the fundamental frequency.
When the fundamental is very high compared to all other frequencies, the planet is tightly bound and considered
very stable as in the cases when the orbital radius is small. However, when the fundamental’s power is low and
proportional to the power of many other frequencies, the planet is loosely bound and is being affected by other
orbital patterns.
On the right is a LogLog plot of the power
of the fundamental frequency versus the
orbital radius at which the planet is
expected to orbit. The expected
P( f f ) ~ 1 / r 2
behavior is seen when r is small. If this
were simply a body orbiting perfectly
about another massive body, this trend
would hold for all r. Here, however, the
pattern becomes significantly disturbed at
a radius of approximately 8A.U..
However, I would even extend this to
make a conjecture that in order to have an
orbit that is guaranteed to be stable, the
orbital radius must be less than
[1.5,2]A.U. since this seems to be the
range at which power deviates from its
expected slope of -2 to something slightly
more shallow, the first indication that this
planet is not orbiting a single free massive
body. This estimate actually agrees with
what is commonly accepted as the critical
radius of orbit.1
The above analyses all use motion in retrograde. It is possible that motion in the prograde direction could
have different stability conditions. One argument for this is that in retrograde, the planet has shorter ‘encounter’
times with the opposing star.
1
http://www.goingfaster.com/icarus/destinationalpha.htm ,
The two graphs above determine more exactly the critical orbital radius for stability, found now to be approximately
2A.U., show that the critical radius is independent of the grade of the orbit, and finally show the stability of
retrograde and prograde orbits side by side. The peaks in the prograde orbits are due to odd resonances with the
second star that cause the entire spectrum to shift very high due to the chaos of the orbit. For example, R = 5.5 in
prograde is the highest peak in the above graph:
After the first 80 year revolution of the binary system, the planet switches stars. The encounter time, the time that
the planet is considered ‘close’ to the second star in its relative orbit is larger in prograde and thus the effect of the
second star is magnified, causing this obvious case’s instability earlier than we saw in retrograde.
Weakly Bound Orbits about the Binary System
The same exact analysis can be performed for orbits with centers corresponding to the center of mass of the
binary star. Again, perfectly circular orbits (achieved with r   and initial conditions ( X p , Y p ) 0  (r ,0) ,

M 1  M 2 A.U .3 
. The main difference between these two classes of orbits is that
(V X p , VY p ) 0   0,2
2 

r
yr


2
one will find the 1 r dependence for the power of the frequency at large r instead of small and deviations from
this at small r instead of large. Also, for the same reason as in tightly bound orbits, prograde will be slightly less
stable than retrograde due to slightly extended ‘encounter’ times with the stars. The code is written to run
simulations of these easily, but none are included here in the interest of brevity and redundancy. The discussion of
weakly bound orbits would match almost exactly that of tightly bound orbits except for the difference mentioned
above.
Other Oddities
As before, I wanted to solve for the position of the stars by inverting the eccentric anomaly equation
T   
a 3
k
   sin  
for  (T ) rather than integrating the positions of the stars in time. A version of the code that uses this method rather
than integrating the stars was created and is attached. However, it does not give the same output as integrating the
function does. For example, this is the output for the 2A.U. case discussed above:
which differs significantly from the integrated form. I spent a lot of time with this and don’t entirely understand the
source other than stating there are inaccuracies that are unaccounted for. Perhaps in calculating the position of the
stars with respect to time more uncertainty arises from each individual position statement (originally stemming from
uncertainty in the Newton’s Approximation of the Root. However, this same result occurs when the tolerance is set
to an absurdly low 1e-12) than integrating it all at once and having error grow slowly with time. Perhaps there is a
time step assumption I’m unaware of and the position(time) calculation of the binary star is one step ahead of the
time… or a combination…
I must admit I’m stumped.