References:Perryman, M. (2011). The exoplanet handbook. Cambridge: Cambridge University Press.Chasnov, J. (2012). Numerical Methods. The Hong Kong University of Science and Technology

Conclusion:From the data I and others in the research group collected, I can conclude:▪ The binary star system itself is stable even when the exo-

planet is added.▪ P-Type orbits are more stable than S-Type orbits▪ P-Type orbits are less susceptible to small changes in initial

conditions.▪ S-Type orbits quickly become unstable when the initial

orbital plane is at an angle greater than 45 degrees.Limitations:• To make the simulator simpler a major assumption has been

made: the binary stars and exoplanets can be modelled as particles, which in actuality is not the case as the closer the bodies get to each other the greater the effects of tidal forces.

• The use of numerical methods also limits the accuracy of this experiment, as there is likely to be an uncertainty in the values the simulator supplies. Since the only data we have is from the simulator it is impossible to determine how large the uncertainty is, but it is likely to be large especially when the speed of the objects is large such as when they are on their closest approach.

Abstract:I will, using a computer simulation of Newton's law of gravity, investigate how the stability of binary star exo-planet orbits are affected by changes in initial conditions.Stability will be determined by analyzing how the planets motion and energies develop over long time scales (Hundreds of Years or as many as can be generated in a limited time frame).I will analyze this data in terms of short and long term trends using properties of these trends such as magnitude and frequency to determine the causes.I will vary different parameters including orbit radius, orbit speed, orbit direction, orbit angle, type of orbit (P or S),... to determine what effect this has on the orbits stability attempting to explain these effects.

Figure 1

The Stability of Binary Star – Exo Planet SystemsJamie Priestley | Dr I Udall | Loreto College

The Binary Star Orbit:The binary stars orbits are highly stable as is to be expected from a symmetrical two mass system, taking the form of interlocking ellipse's with the barycentre as a focal point or in special cases a circle centred on the barycentre. The speed of the stars varies being maximum at the closest approach.

The Semi Major Axis:As total energy of the system increases the length of the semi major axis increases.This is achieved by:• Decreasing Mass• Increasing Velocity• Increasing Separation

The Semi Minor Axis:As total energy of the system increases the length of the semi minor axis increases.This is achieved by:• Decreasing Mass• Increasing Velocity• Increasing Separation

Escape Velocity:Escape velocity is achieved when the total energy of the system is positive (the magnitude of the kinetic energy is greater than the magnitude of the gravitational potential energy). When the escape velocity is reached the star may exit the system (as seen in several of the above).

Circular Orbit:If the mass, separation and velocity of the binary stars are balanced a circular orbit can occur. For example when the stars are separated by 20 AU, the mass of each star is 333060 earth masses and the velocity of each star has magnitude of ~1 AU/year and opposite directions the start orbits lie on the same circle centred on the barycentre with radius 5 AU.

The simulator consists of a set of 2nd order differential equations, these equations cannot be solved analytically in the general case, so an approximation must be calculated using an appropriate numerical method.

The Euler method:The Euler method works by calculating the value of the gradient at a

given point then using this to make a small step to the next point

along, the smaller the step the more accurate the approximation as

can be seen Figure 2 where the red line represents the real function,

the blue and green lines the Euler method with step sizes of 1 and

0.25 respectively.

ℎ = 𝑠𝑡𝑒𝑝 𝑠𝑖𝑧𝑒

𝑓 𝑥, 𝑦 =𝑑𝑦

𝑑𝑥|𝑥,𝑦

𝑥𝑛+1 = 𝑥𝑛 + ℎ

𝑦𝑛+1 = 𝑦𝑛 + ℎ𝑓(𝑥𝑛, 𝑦𝑛)

Runge-Kutta Method:In order to maximise accuracy the Runge-Kutta method was used which is an improvement over the Euler method. The Runge-Kutta method uses an approximation of the gradient at the middle and end point of each step as well as the gradient at the start point to produce multiple estimates of the new value, these approximations are then averaged to produce a more accurate estimate of the new value (in this case the objects velocity and position).

ℎ = 𝑠𝑡𝑒𝑝 𝑠𝑖𝑧𝑒

𝑓 𝑥, 𝑦 =𝑑𝑦

𝑑𝑥|𝑥,𝑦

𝑥𝑛+1 = 𝑥𝑛 + ℎ𝑘1 = ℎ𝑓(𝑥𝑛, 𝑦𝑛)

𝑘2 = ℎ𝑓(𝑥𝑛 +1

2ℎ, 𝑦𝑛 +

1

2𝑘1)

𝑘3 = ℎ𝑓(𝑥𝑛 +1

2ℎ, 𝑦𝑛 +

1

2𝑘2)

𝑘4 = ℎ𝑓(𝑥𝑛 + ℎ, 𝑦𝑛 + 𝑘3)

𝑦𝑛+1 = 𝑦𝑛 +1

6(𝑘1 + 2𝑘2 + 2𝑘3 + 𝑘4)

Units:Mass – Earth masses

Distances – AUTime - Years

Figure 2

The Equations:The simulator is based upon a set of physics equations which govern the motion of massive bodies.These Include Newton's Law of Gravity, Newton's Third Law, and others.

𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝒓 ×𝑚 ሶ𝒓

𝐹𝑇 = 𝑚𝑎 𝐺𝑃𝐸 = −𝐺𝑀1𝑀2

𝑟

𝐹𝐺 =𝐺𝑀1𝑀2

𝑟2𝐾𝐸 =

1

2𝑚𝑣2

Types of Orbits:There are two types of binary star exo-planet orbits P-type and S-type.

P-type orbits consist of a planet orbiting the whole binary star system.

S-type orbits consist of a planet orbiting just one of the stars following it along its trajectory.

P-type:• More stable than S-type orbit.• Less susceptible to changes in initial conditions than S-type orbit.

Likely because the magnitude of resultant forces is more consistent than in in S-Type orbits and is always pointed approximately towards the barycentre.

Also since the distance between the planet and star is greater the effects of the individual stars on the planet is reduced.

S-type:• Less stable than P-type orbit.• More susceptible to changes in initial conditions P-type orbit.

Instability likely caused by the constantly changing acceleration of the planet. For example when the planet is in-between both stars the acceleration will be small directed towards either star but when the order is star-star-planet the acceleration will be greater and directed towards the stars.

The Energy Pattern:The energy of the exo-planet varies constantly with two main patterns.

A smaller pattern where each of the energies increases and decreases rapidly with time periods on the order of 1 year.

The other larger pattern of increasing and decreasing amplitude has time period on the order of 10 years.

The smaller trend is likely caused by the orbit of the planet and the larger trend the orbit of the stars.

The Trend:As the angle of inclination increases the stability of the exo-planets orbit decreases.

The orbit becomes highly unstable at around 60 degrees with an irregular pattern, and unpredictable spikes in kinetic energy.

If the orbit is inclined at a large enough angle escape velocity is achieved and the planet leaves the system.

Orbital Trajectory Exo-Planet:When the initial orbital plane is inclined, the orbit of the planet relative to the star as shown to the left is chaotic with a constantly changing plane.

Orbital Trajectory Binary Stars:The binary stars orbit is only negligibly effected by the planet in a S-type orbit and is still highly stable.