Welcome to the I Can't Sleep Podcast,
Where I help you drift off one fact at a time.
I'm your host,
Benjamin Boster,
And today's episode is about orbits.
In celestial mechanics,
An orbit is the curved trajectory of an object under the influence of an attracting force.
Alternatively,
It is known as an orbital revolution because it is a rotation around an axis external to the moving body.
Examples for orbits include the trajectory of a planet around a star,
A natural satellite around a planet,
Or an artificial satellite around an object,
Or position in space,
Such as a planet,
Moon,
Asteroid,
Or Lagrange point.
Normally,
Orbit refers to a regularly repeating trajectory,
Although it may also refer to a non-repeating trajectory.
To a close approximation,
Planets and satellites follow elliptic orbits,
With the center of mass being orbited at a focal point of the ellipse,
As described by Kepler's laws of planetary motion.
For most situations,
Orbital motion is adequately approximated by Newtonian mechanics,
Which explains gravity as a force obeying an inverse square law.
However,
Albert Einstein's general theory of relativity,
Which accounts for gravity as due to curvature of spacetime,
With orbits following geodesics,
Provides a more accurate calculation and understanding of the exact mechanics of orbital motion.
Historically,
The apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres.
This model positioned the existence of perfect moving spheres,
Or rings,
To which the stars and planets were attached.
It assumed the heavens were fixed,
Apart from the motion of the spheres,
And was developed without any understanding of gravity.
This concept originated with Hellenistic astronomy,
Particularly Eudoxus and Aristotle.
After the planets' motions were more accurately measured,
Theoretical mechanisms,
Such as deferent and epicycles,
Were added by Ptolemy.
Although the model was capable of reasonably accurately predicting the planets' positions in the sky,
More and more epicycles were required as the measurements became more accurate.
Hence,
The model became increasingly unwieldy.
Originally geocentric,
It was modified by Copernicus to place the Sun at the center,
To help simplify the model.
The model was further challenged during the 16th century,
As comets were observed traversing the spheres.
The basis for the modern description of orbits was first formulated by Johannes Kepler,
Whose results are summarized in his Three Laws of Planetary Motion.
First,
He found that the orbits of the planets in the solar system are elliptical,
Not circular,
Or epicyclic,
As had previously been believed,
And that the Sun is not located at the center of the orbits,
But rather at one focus.
Second,
He found that the orbital speed of each planet is not constant,
As had previously been thought,
But rather that the speed depends on the planet's distance from the Sun.
Third,
Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun.
For the planets,
The cubes of their distances from the Sun are proportional to the squares of their orbital periods.
Jupiter and Venus,
For example,
Are respectively about 5.
2 and 0.
723 astronomical units distant from the Sun.
Their orbital periods respectively about 11.
86 and 0.
615 years.
The proportionality is seen by the fact that the ratio for Jupiter,
5.
204 cubed divided by 11.
862 squared,
Is approximately equal to 1.
002,
Is practically equal to that for Venus.
0.
723 cubed divided by 0.
615 squared is approximately equal to 0.
999,
In accord with the relationship.
Idealized orbits meeting these rules are known as Kepler orbits.
Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation,
And that in general the orbits of bodies subject to gravity were conic sections,
Under his assumption that the force of gravity propagates instantaneously.
To satisfy Kepler's third law,
Newton showed that for a pair of bodies,
The orbit size,
A,
Orbital period,
T,
And their combined masses,
M,
Are related to each other by T squared is proportional to A cubed over M,
And that those bodies orbit their common center of mass.
Where one body is much more massive than the other,
As is the case of an artificial satellite orbiting a planet,
It is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.
Advances in Newtonian mechanics were then used to explore variations from the simple assumptions behind Kepler orbits,
Such as the perturbations due to other bodies,
Or the impact of spheroidal rather than spherical bodies.
Joseph-Louis Lagrange developed a new approach to Newtonian mechanics,
Emphasizing energy more than force,
And made progress on the three-body problem,
Discovering the Lagrangian points with Euler.
In a dramatic vindication of classical mechanics,
In 1846,
Urbain Le Verret was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus.
Albert Einstein,
In his 1916 paper The Foundation of the General Theory of Relativity,
Explained that gravity was due to curvature of space-time,
And removed Newton's assumption that changes in gravity propagate instantaneously.
This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits.
In relativity theory,
Orbits follow geodesic trajectories,
Which are usually approximated very well by the Newtonian predictions,
Except where there are very strong gravity fields and very high speeds,
But the differences are measurable.
Essentially,
All the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy.
The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion.
First noted by Le Verret.
However,
Newton's solution is still used for most short-term purposes,
Since it is significantly easier to use and sufficiently accurate.
Within a planetary system,
Various non-stellar objects follow elliptical orbits around the system's barycenter.
These objects include planets,
Dwarf planets,
Asteroids,
And other minor planets,
Comets,
Meteoroids,
And even space debris.
A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system.
Bodies that are gravitationally bound to one of the planets in a planetary system,
Including natural satellites,
Artificial satellites,
And the objects within ring systems,
Follow orbits about a barycenter near or within that planet.
Owing to mutual gravitational perturbations,
The eccentricities and inclinations of the planetary orbits vary over time.
Mercury,
The smallest planet in the solar system,
Has the most eccentric orbit.
At the present epoch,
Mars has the next largest eccentricity,
While the smallest orbital eccentricities are seen with Venus and Neptune.
As two objects orbit each other,
The periapsis is that point at which the two objects are closest to each other.
Less properly,
Perifocus or pericentron are used.
The apoapsis is that point at which they are the farthest,
Or sometimes apofocus or apocentron.
A line drawn from periapsis to apoapsis is the line of apsides.
This is the major axis of the ellipse,
The line through its longest part.
More specific terms are used for specific bodies.
For example,
Perigee and apogee are the lowest and highest parts of an orbit around Earth,
While perihelion and aphelion are the closest and farthest points of an orbit around the Sun.
Things orbiting the Moon have a perilune and apolune,
Or perisaline and aposaline respectively,
And orbit around any star,
Not just the Sun,
As a periastron and apastron.
In the case of planets orbiting a star,
The mass of the star and all its satellites are calculated to be at a single point called the barycenter.
The individual satellites of that star follow their own elliptical orbits with the barycenter at one focal point of that ellipse.
At any point along its orbit,
Any satellite will have a certain value of kinetic and potential energy with respect to the barycenter,
And the sum of those two energies is a constant value at every point along its orbit.
As a result,
As a planet approaches periapsis,
The planet will increase in speed as its potential energy decreases.
As a planet approaches apoapsis,
Its velocity will decrease as its potential energy increases.
An orbit can be explained by combining Newton's laws of motion with his law of universal gravitation.
The laws of motion are as follows.
A body continues in a state of uniform rest,
Or motion,
Unless acted upon by an external force.
The acceleration produced when a force acts is directly proportional to the force and takes place in the direction in which the force acts.
To every action,
There is an equal and opposite reaction.
By the first law of motion,
In the absence of gravity,
A physical object will continue to move in a straight line due to inertia.
According to the second law,
A force,
Such as gravity,
Pulls the moving object toward the body that is the source of the force and thus causes the object to follow a curved trajectory.
If the object has enough tangential velocity,
It will not fall into the gravitational body but can instead continue to follow the curved trajectory caused by the force indefinitely.
The object is then said to be orbiting the body.
According to the third law,
Each body applies an equal force on the other,
Which means the two bodies orbit around their center of mass or barycenter.
Because of the law of universal gravitation,
The strength of the gravitational force depends on the masses of the two bodies and their separation.
As the gravity varies over the course of the orbit,
It reproduces Kepler's laws of planetary motion.
Depending on the evolving energy state of the system,
The velocity relationship of two moving objects with mass can be considered in four practical classes,
With subtypes.
No-orbit,
Suborbital trajectories,
A range of interrupted elliptical paths,
Orbital trajectories or simply orbits,
Range of elliptical paths with closest point opposite firing point,
Circular path,
Range of elliptical paths with closest point at firing point,
Open or escape trajectories,
Parabolic paths,
Hyperbolic paths,
Parallel paths.
To achieve orbit,
Conventional rockets are launched vertically at first to lift the rocket above the dense lower atmosphere,
Which causes frictional drag,
And gradually pitch over and finish firing the rocket engine parallel to the atmosphere to achieve orbital injection.
Once in orbit,
Their speed keeps them above the atmosphere.
If an elliptical orbit dips into dense air,
The object will lose speed and re-enter,
Falling to the ground.
Occasionally,
A spacecraft will intentionally intercept the atmosphere in an act commonly referred to as an arrow-breaking maneuver.
As an illustration of an orbit around a planet,
The Newton's cannonball model may prove useful.
This is a thought experiment,
In which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle speed.
The effects of air friction on the cannonball are ignored,
Or perhaps the mountain is high enough that the cannon is above the Earth's atmosphere,
Which is the same thing.
If the cannon fires its ball with a low initial speed,
The trajectory of the ball curves downward and hits the ground.
As the firing speed is increased,
The cannonball hits the ground farther away from the cannon,
Because while the ball is still falling towards the ground,
The ground increasingly curving away from it.
All these motions are actually orbits in a technical sense.
They are describing a portion of an elliptical path around the center of gravity,
But the orbits are interrupted by striking the Earth.
If the cannonball is fired with sufficient speed,
The ground curves away from the ball at least as much as the ball falls,
So the ball never strikes the ground.
It is now in what could be called a non-interrupted or circumnavigating orbit.
For any specific combination of height above the center of gravity and mass of the planet,
There is one specific firing speed,
Unaffected by the mass of the ball,
Which is assumed to be very small relative to the Earth's mass,
That produces a circular orbit.
As the firing speed is increased beyond this,
Non-interrupted elliptic orbits are produced.
If the initial firing is above the surface of the Earth,
There will also be non-interrupted elliptical orbits at slower firing speed.
These will come closest to the Earth at the point half an orbit beyond and directly opposite the firing point below the circular orbit.
At a specific horizontal firing speed called escape velocity,
Dependent on the mass of the planet and the distance of the object from the barycenter,
An open orbit is achieved that has a parabolic path.
At even greater speeds,
The object will follow a range of hyperbolic trajectories.
In a practical sense,
Both of these trajectory types mean the object is breaking free of the planet's gravity and going off into space,
Potentially never to return.
However,
The object remains under the influence of the Sun's gravity.
In most real-world situations,
Newton's laws provide a reasonably accurate description of motion of objects in a gravitational field.
The adjustments needed to accommodate the theory of relativity become appreciable in cases where the object is in the proximity of a significant gravitational source,
Such as a star,
Or a high level of accuracy is needed.
The acceleration of a body is equal to the combination of the forces acting on it,
Divided by its mass.
The gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them.
For a two-body problem,
Defined as an isolated system of two spherical bodies with known masses and sufficient separation,
This Newtonian approximation of their gravitational interaction can provide a reasonably accurate calculation of their trajectories.
If the heavier body is much more massive than the smaller,
As in the case of a satellite or small moon orbiting a planet,
Or for the Earth orbiting the Sun,
It is accurate enough and convenient to describe the motion in terms of a coordinate system that is centered on the heavier body.
And we say that the lighter body is in orbit around the heavier.
For the case where the masses of two bodies are comparable,
An exact Newtonian solution is still sufficient and can be had by placing the coordinate system at the center of the mass of the system.
Energy is associated with gravitational fields.
A stationary body,
Far from another,
Can do external work if it is pulled towards it and therefore has gravitational potential energy.
Since work is required to separate two bodies against the pull of gravity,
Their gravitational potential energy increases as they are separated,
And decreases as they approach one another.
For point masses,
The gravitational energy decreases to zero as they approach zero separation.
It is convenient and conventional to assign the potential energies having zero value when they are an infinite distance apart.
And hence it has a negative value since it decreases from zero for smaller finite distances.
When only two gravitational bodies interact,
Their orbits follow a conic section.
The orbit can be open,
Implying the object never returns,
Or closed,
Returning.
Which it is depends on the total energy,
Kinetic plus potential energy of the system.
In the case of an open orbit,
The speed at any position of the orbit is at least the escape velocity for that position.
In the case of a closed orbit,
The speed is always less than the escape velocity.
Since the kinetic energy is never negative if the common convention is adopted of taking the potential energy as zero at infinite separation,
The bound orbits will have negative total energy,
The parabolic trajectories zero total energy,
And hyperbolic orbits positive total energy.
An orbit will have a parabolic shape if it has a velocity of exactly the escape velocity of that point in its trajectory.
And it will have the shape of a hyperbola when its velocity is greater than the escape velocity.
When two bodies approach each other with escape velocity,
Or greater relative to each other,
They will briefly curve around each other at the time of their closest approach,
And then separate and fly apart.
All closed orbits have the shape of an ellipse.
A circular orbit is a special case where the foci of the ellipse coincide.
Bodies following closed orbits repeat their paths with a certain time called the period.
This motion is described by the empirical laws of Kepler,
Which can be mathematically derived from Newton's laws.
These can be formulated as follows.
One,
The orbit of a planet around the sun is an ellipse with the sun in one of the focal points of that ellipse.
This focal point is actually the barycenter of the sun-planet system.
For simplicity,
This explanation assumes the sun's mass is infinitely larger than the planet's.
The planet's orbit lies in a plane called the orbital plane.
Two,
As the planet moves in its orbit,
The line from the sun to the planet sweeps a constant area of the orbital plane for a given period of time,
Regardless of which law of its orbit the planet traces during that period of time.
This means that the planet moves faster near its perihelion than near its apohelion.
Because,
At the smaller distance,
It needs to trace a greater arc to cover the same area.
This law is usually stated as equal areas in equal time.
Three,
For a given orbit,
The ratio of the cube of its semi-major axis to the square of its period is constant.
Ideally,
The bound orbits of a point mass or a spherical body with a Newtonian gravitational field form closed ellipses,
Which repeat the same path exactly and indefinitely.
However,
Any non-spherical or non-Newtonian effects will cause the orbit's shape to depart from the ellipse.
Such effects can be caused by a slight oblateness of the body,
Mass anomalies,
Tidal deformations,
Or relativistic effects,
Thereby changing the gravitational field's behavior with distance.
The two-body solutions were published by Newton in Principia in 1687.
In 1912,
Carl Frithjof Sundman developed a converging infinite series that solves the general three-body orbit.
However,
It converges too slowly to be of much use.
The restricted three-body problem in which the third body is assumed to have negligible mass has been extensively studied.
The solutions to this case include the Lagrangian points.
In the case of lunar theory,
The 19th century work of Charles-Eugène Doulani allowed the motions of the moon to be predicted to within its own diameter over a 20-year period.
No universally valid method is known to solve the equations of motion for a system with four or more bodies.
The following derivation applies to an elliptical orbit.
The assumption is that the central body is massive enough that it can be considered to be stationary,
And so the more subtle effects of general relativity can be ignored.
The Newtonian law of gravitation states that the gravitational acceleration of the second mass towards the central body is related to the inverse of the square of the distance between them.
Namely,
The force on object 2 equals negative g times m,
1 times m,
2 over r-squared,
Where F-sub-2 is the force acting on the mass m-sub-2 caused by the gravitational attraction mass m-sub-1 as for m-sub-2.
G is a universal gravitational constant,
And r is the distance between the two masses' centers.
From Newton's second law,
The summation of the forces acting on m-sub-2 related to that body's acceleration,
The force on object 2 equals its mass times its acceleration,
Where a-sub-2 is the acceleration of m-sub-2 caused by the force of gravitational attraction F-sub-2 of m-sub-1 acting on m-sub-2.
