Planetary Motion

The Universal Law of Gravitation states that every body in this universe attracts the other by virtue of its mass, The force of attraction being directly proportional to the product of their masses and inversely proportional to the square of the distance of their separation. This law was first published by Sir Isaac Newton in 1687.







'Restart' starts the simulation using the previously-entered parameters.
'Simulate' starts the simulation using the currently-entered parameters.

Star

Mass: X coordinate: Y coordinate: Horizontal velocity: Vertical velocity:

Planet 1

Mass: X coordinate: Y coordinate: Horizontal velocity: Vertical velocity:

Planetary Phenomena

  1. Unstable orbit: Here, the planet's total energy is actually positive. Hence it is not actually bound to the star and instead revolves around an unstable orbit before getting ejected. Moreover, the planet is only 10 times lighter than the star, causing the star to wobble.


  2. Two planetary orbits: Two planets revolving around a star in remarkably stable orbits.


  3. Binary system: Two equally heavy bodies orbiting their center of mass. This sort of behaviour is usually seen among stars. Alpha Centauri A and B are well-known examples, just 4.36 light years away from Earth. However, another star named Proxima Centauri actually orbits the binary system!


  4. Intersecting orbits: Two planets revolving around a star in intersecting orbits.


  5. Planetary dance: The two planets initially revolve symmetrically around the star. However, they come quite close four times and exchange energy during each interaction. The planet that continually loses energy occupies a lower orbit and eventually gets ejected whereas the planet that continually gains energy settles in an highly elliptical orbit.



Description

The Force of Gravity: \[ F_g = \frac{G m_1 m_2}{r^2} \]
  1. As a result of Newton's 3rd law of motion, the gravitational force of attraction of body A on body B is equal to that of body B on body A.
  2. G is an universal constant whose value is known to be 6.67 * 10-11 m3kg-1s-2
  3. As stated before, gravitational force acts between all pairs of objects in this universe by virtue of their mass. However, we don't see everyday objects getting attracted to each other because the product of their masses is not high enough to dwarf the low value of G. Hence, the force of gravity between them is too low to cause any observable effects.
  4. Although it is a well-known fact that electromagnetic force is 1038 times stronger than gravitational force, electromagnetic force cannot be dominant in the universe due to the presence of two types of charge. Over large volumes, these two types of charges cancel out each other, thereby nullifying any electromagnetic interactions. Moreover, electromagnetic interactions can be attractive as well as repulsive. On the other hand, there is only one type of mass, hence there is no repulsive gravitational interaction and no mass cancellation throughout the entire expanse of the universe, thereby allowing gravity to pervade it.
  5. The motion of celestial bodies is directed by the force of gravity alone. Their huge masses allows high degree of gravitational interaction despite the large distances between them.
The Gravitational Field: \[ E (j) = \sum_i \frac{G m_i}{r_{ij}^2} \] \[ F_g = m E(j) \]
  1. Physicists often use the concept of a gravitational field being generated around a mass. The strength of the gravitational field at a point in the vicinity of a mass is proportional to the mass 'M' of the object and falls as the square of distance of separation 'r'. The above formula generalises the gravitational field at point \(j\) due to \(i\) bodies. \(r_{ij}\) is the distance between them.
  2. Any body of mass 'm', when kept in this field, interacts with it. The strength of interaction being proportional to 'm' and to the strength of the gravitational field.
  3. The postulation of a field is not merely theoretical. Infact, it is known that the bounds of this field explands at the speed of light. Moreover, the concents of fields and potentials convey the transfer of energy and momentum in an elegant manner.
Gravitational Potential \[ V (j) = -\sum_i \frac{G m_i}{r_{ij}} \] \[ U = \sum_{i \not = j} \frac{G m_i m_j}{r_{ij}} \]
  1. When two bodies are gravitationally interacting with one another, some energy is required to separate them until the gravitational force of attraction between them becomes negligible. This energy, but for a unit mass, is defined to be the gravitational potential.
  2. The gravitational potential \(V\) at a point 'r' units away from a mass 'M' is defined to be -GM/r whereas the potential energy of a mass 'm' placed at this point is defined to be -GMm/r
  3. The negative sign denotes that 1) energy is required to separate them or 2) the bodies are bound to each other
  4. The above formula generalises gravitational potential and potential energy for multiple bodies
  5. The total mechanical energy as a result of gravitational interaction between two objects is equal to the sum of their kinetic energy and gravitational potential energy.
Kepler's Laws
  1. In the 17th century, Johannes Kepler postulated three laws that were followed by orbiting planets. These laws were obtained by analysis of planetary data recorded by Tycho Brahe with the assistance of Kepler. They lacked any scientific backing when they were proposed.
  2. The first law states that planets revolve around their star in elliptical orbits, wherein the star is situated at one of the foci. An ellipse is the locus of a point such that the sum of their distance from two fixed points is always constant. In theory, a circle is an ellipse obtained by a merger of these two points. This law is a result of the nature of gravitational force itself.
  3. The second law states that planets trace equal areas in equal intervals of time. This indirectly means that planets in non-circular orbits must move faster when closer to the star. This can be observed in the default simulation above. This law is a result of conservation of angular momentum
  4. The third law states that the square of time period of revolution of a planet around its star is directly proportional to the cube of the major axis. This law can be derived with a bit of algebra.
General info about stellar systems
  1. It is a false notion that all planets revolve around their star. The planets and the star revolve around the center of mass of the stellar system. Due to the fact that stars tend to be much more massive than all the revolving planets combined, this center of mass often tends to be within the star and maybe very close to the center of the star, therefore giving the appearance that the planets actually revolve around the star.
  2. Planets are usually formed from a disk of rocks and gases found around the protostar. Since a disk is planar, all the resulting planets tend to orbit the star in the same plane, with small deviations.
The Three Body Problem and Chaos
  1. While Newton's laws perfectly describe the behaviour of two gravitationally attracted bodies, describing the motion of three bodies turned out to be elusive.
  2. It was also noticed that a small change to the initial coordinates of velocites of a planet (in three or multi-body motion) can have a drastic change in the long-term behaviour of the system. This phenomenon is known as chaos.


Note:
  1. The above simulation is not compatible with SI units. The value of G is taken to be 1, in order to avoid working with huge masses and astronomical distances
  2. Stars and Planets in the above simulation are treated equally. This allows us to observe the wobbling of the star when the revolving planets are not many orders lighter than the star.
  3. The above simulation doesn't check and allow collissions between bodies. Because of this, ejection of planets is a very common occurence in the simulation although it rarely takes place in reality.
  4. Additionally, the simulation uses Euler method to solve for the trajectory of the planet. This is just a first order method, and is prone to errors.
  5. Related: Spring Motion, Projectile Motion, Double Pendulum, Lorenz System

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