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Introduction

In this blog post, I am going to explain general knowledge about mechanics of Planetary Motion. But before starting this, you should have at least a loose idea of Vectors and few other very simple things. I have written few formulas but there is no derivation here. What is stressed here is meaning of those formulas rather than mathematics. For more details, you can check out the references provided at the end.

Newton’s Gravitation and Circular Orbits

In 17th century, Sir Isaac Newton explained than every object thrown upward in air has tendency to fall back on floor due to a mysterious force which he called Gravity. He said that every two objects which possess some mass and are kept at certain distance, exert this force on each other. This force is directly proportional to product of their masses and inversely proportional to square of distance between them. It is an especially important result which we now call as Newton’s Gravitation Law. Mathematically, it written as

Fig. 1
G = is gravitation constant

(Here onwards, this system of mass m and M will be used to illustrate things)

When a force acts on a body it causes the body to accelerate which results in change of velocity. Because velocity is a vector, it changes when any or both of its magnitude and direction changes (a single force can result in a change of both). Component of force that cause change in magnitude is called tangential acceleration and the component that changes direction is called centripetal or normal force.

When a body moves in a circular path with uniform speed, only direction of velocity changes due to centripetal forces (which acts toward center of circle). This force maintains the circular path. For example, if a planet moves in circular orbit around another, gravitation between them provides that centripetal force to sustain circular orbit.

(RHS of this equation is formula of centripetal force, acting on a body moving with speed v, on a circular path of radius r)

Fig. 2

Gravitation Potential Energy

Potential energy is energy of a body due to its position (or configuration of system) in certain field of force. Generally, potential energy does not have an absolute value, but relative value (more proper concept is Potential difference). Obviously, Gravitation does not act through direct contact of bodies. It acts through an invisible gravitational field. When a body is placed in a gravitational field of another body (distance between them is r), then its potential energy is given by the formula

(This formula is based on the assumption that potential energy of m, when kept at infinite distance from M is zero)

This negative sign here is very important because it tells that when the bodies brought closer from infinite distance, the potential energy decreases from zero to negative values. Thus, the closer they are, smaller potential energy will be.

Polar Coordinates and Angular Momentum

A coordinate system is a very useful and practical way of defining position of a body. There are different types of coordinate system. The most common one is a Cartesian coordinate system, however 2-dimensional polar coordinate system is more useful in this case. In 2D polar coordinate system, the position of a point is described by defining a position vector (r) from a certain reference point. The length r of this vector describes distance from reference point and direction is described by the angle θ which it makes from a known reference line (see fig. 5). Let’s say this particle shown in figure has certain velocity. Here, this velocity vector (v) at this instant can be broken into two components namely vr and vθ (as shown in figure). This vr causes change in length r of position vector and vθ causes change in angle θ. In case of circular motion, only vθ exists.

Fig. 3
Comparison of Grids between Cartesian and Polar Coordinate System.
Fig. 4
How Position Vectors and corresponding unit vectors appear in polar coordinates

Fig. 5
How velocity vector is broken into components in polar coordinates

Angular Momentum is another property measured for a body in motion. Fundamentally, it is measured with respect to (w.r.t.) a point in space. Generally, this point is same as previous reference point from where position vector is defined. Mathematically, it is cross product of position and momentum vector.

As you can see, Angular momentum is itself a vector. Its magnitude is

Fig. 6
Direction of Angular Momentum Vector

Direction is perpendicular to both position and momentum vector. Now as force causes a change in momentum, change in angular momentum is caused by Moment of Force, which is nothing but cross product of position and force vector. Mathematically, it is

If you choose a reference point such that moment of force w.r.t. that point vanishes, then angular momentum w.r.t. that point will remain constant. Luckily in our gravitation problem such point is center of body with mass M, because Gravitational Force and position vector from there become anti-parallel.

Thus, angular momentum of body of mass m w.r.t. center of body with mass M is constant. Constancy of Angular momentum vector assures that mass m will move in a plane.

Possible Shapes of Orbits

If two bodies are kept near each another, without any initial velocity, then due to gravitation they will be moving toward each other, rectilinearly, along the line joining them. But in case if one of them (mass m) enters in gravitation field of another (mass M), then different trajectories are possible.

In case if initial velocity vector of mass m is directed towards the center of mass M, then it will again perform rectilinear motion. But if it is directed anywhere except the center of mass M, it will move in a plane and shape of trajectory can be any conic section, i.e., Circle, Ellipse, Parabola, and Hyperbola.

This result can also easily be derived by using calculus.

Kinetic Energy and Centrifugal Potential

Kinetic energy is energy of a body which it possesses due to its state of motion. Mathematically, it is given by

Where, v is velocity of mass m. Now this v2 can be expressed as sum of vi2 where I represent different independent dimension (or orthogonal components of velocity vector). It means kinetic energy can be expressed as sum of kinetic energy due to motion along different dimensions.

Like in previous case where v was broken into vr and vθ, then total kinetic energy is sum of kinetic energy due to vr (Kr) and that due to vθ (Kθ). In equations, it is

Now, due to constancy of angular momentum, component of kinetic energy due to vθ can be expressed as function of r, i.e., it varies with change of r only (and r varies due to vr). The mathematical relation is of inverse square.

Because Kθ is function of its position (r), so it behaves like potential energy (or potential) and thus it is treated like a potential function in this case. Here, this Kθ is called Centrifugal Potential.

The sum of Centrifugal potential (Ucen) and Gravitation potential give us Effective potential (Ueff), which off course is function of position (r). The graph of effective potential energy like as shown in figure 7.


Fig. 7
Graph of Effective Potential

As always, total mechanical energy is sum of kinetic and potential energy. So is here

Kr is kinetic energy due to vr. Because gravitational field is conservative in nature, total mechanical energy does not change during motion under gravitational field.

As previously mentioned, when m enters the fields of M with certain velocity which is not directing directly toward center of M, it moves along a plane on a path of shape of a conic sections (ellipse, circle, etc). There is a quantity called eccentricity (e) which defines which conic section it is. Here in this motion, that eccentricity is a function of total energy. That is why the shape of trajectory depends on the total energy with which mass m enters in the field of mass M.

Here onwards, there will be 4 different cases:

  1. Case 1: When E > 0, then e > 1, which means shape of orbit will be a Hyperbola. That is, it comes from infinity, approaches M up to a certain distance then returns. (You can see in graph that in case 1, r is lower bound, i.e., it cannot have value less than a certain value because there will be not enough kinetic energy (Kr) to convert into Ueff in order to get closer).
  2. Case 2: When E = 0, then e = 1, which means shape of orbit is Parabola. It is somewhat similar to previous case, except the fact that parabola and hyperbola have different shapes thus different properties. Otherwise here also r is lower bound.
  3. Case 3: When Emin < E < 0, then 0 < e < 1, that is, it is an Ellipse. Now here r is lower bound as well as upper bound, means here the value of r will be between two fixed values which is case with ellipse. (You can again see in graph, that in case 3, in order to get closer than minimum value of separation or farther than maximum value, Ueff needs to increase and again there won’t be sufficient Kr for this.
  4. Case 4: When E = Emin, then e = 0, that is, a Circle. Now there, Kr = 0, i.e., no motion along line joining the two (radius), thus separation remains the same.

(All possible orbits for m around M is shown in fig. 8)

Fig. 8
Possible Orbits of m around M

Conclusion

Planets moving under the influence of other’s gravity can move in path with various shapes, depending on how it enters in that gravitational field and with how much energy. It is due to inverse square relation of gravitation force in Newton’s Gravitation Law, that the trajectories have shape of conic sections. It is total energy of the moving planet or body which determines of which shape the path will be.

Written by: Sara Nadzak and Rohit Verma

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