How do the angular momentum and orbital velocity affect the eccentricities of planetary orbits?

This essay's major purpose is to provide an explanation for the research question, "How do the Angular Momentum and Orbital Velocity Affect the Eccentricity of Planetary Orbits?" This study establishes this relationship using Kepler's laws and celestial mechanics. Also, the essay has used planetary data to corroborate the thesis in order to give it a quantitative dimension. The solar system has provided the data, which has been gathered after countless tests.
The study of organic and man-made entities in space is known as celestial mechanics1. The initial intention was to deduce Kepler's Laws from Newton's syntheses; however, the research was expanded as eccentricity and other characteristics were incorporated. Reading a significant work in the astrophysics magazine that examined "the impact of abrupt mass loss and a random kick velocity caused by a supernova explosion on the dynamics of a binary star" also aroused my interest in this subject. The article concentrated on how several parameters, including the semi-major axis and the periastron lengths, were affected by mass loss and random kick velocity. Several of those parameters were compatible with the eccentricity of planetary orbits, which gave rise to the current study subject. The initial intention was to deduce Kepler's Laws from Newton's syntheses; however, the research was expanded as eccentricity and other characteristics were incorporated.

 

Celestial mechanics theories have been discussed since 340 BC. Yet it wasn't until Sir Issac Newton codified the Law of Gravitation and elucidated the Johannes Kepler Laws that the key advancement was made (1571-1630) 2. Three planetary motion rules were first proposed by German mathematicians Astronomia Nova and Harmonices Mundi, and these laws later served as the foundation for keplerian dynamics. "New Astronomy Based upon Causes, or Celestial Physics, Treated by Means of Commentary on the Motion of the Star Mars, from Observations of Tycho Brahe, Gent." is the entire title in translation. Although Astronomia Nova and Harmonices Mundi made for captivating reading, these principles at the time were unproven due to a lack of mathematical understanding and the use of concepts that weren't found until much later. Kepler was also unable to explain why his rules held true since he rejected the notion that magnetic forces were responsible for eccentric orbits. Yet, after the creation of Newton's laws, Kepler's laws were also confirmed because they could be mathematically explained if the laws of motion and gravitational attraction were real. After the discovery of Kepler's Laws, the study of celestial mechanics expanded to include the study of space flight, which was one of its practical applications. Although Celestial Mechanics has been extensively researched throughout history, this does not lessen the need for more investigation because there is no end to discovery. Celestial mechanics has developed into a significant foundation for space travel and other subjects that have branched off of it, such as orbital determination, which has proven essential to understanding exo-planets.

The path a planet takes around a primary body is referred to as an orbit. During the early stages of the development of information about planetary motion, it was thought that magnetic forces were to blame for this course. Kepler's rules, however, could not be reconciled with this notion. Philosophiae Naturalis Principia Mathematica, which Sir Isaac Newton published in 1687, is the most significant work in the physical sciences. According to Newtonian mechanics, the central mass's gravity is what initially shapes the trajectory. The force between any two bodies known as gravity is known to be directly proportional to the product of the masses of the bodies and inversely proportional to the squares of their distances, according to Newtonian Mechanics. Gravitation was commonly understood as an instantaneous force, and other interpretations built on this notion. The concept of gravity, however, changed from being a force to being a distortion in spacetime caused by the mass of the central body following great progress in the study of general relativity.

 

Together with other variables, gravity essentially determines the features of planetary orbit. The orbits can then be divided into different categories. These divisions are based on the orbit geometry and the core objects that the body revolves around. The classification of orbits based on eccentricity is the main topic of this essay. Given by is the Orbital Equation.

 

r = \(\frac{c}{1-𝜀\ 𝑐𝑜𝑠θ}\) (1.0)

c = \(\frac{J^2}{GMm^2}\) (1.1)

Eccentricity measures how far the closed figure deviates from a circle. The shapes that are thought of as conic sections can be used to quantify eccentricity. A circle has an eccentricity of 0, and the value rises as the shape becomes flatter. The shape's centre progressively moves away from its original centre as well. Typically, eccentricity is indicated by the symbol " ‘𝜀," as it is in the orbit equation (1.2). By calculating the ratio between the length of the semi-major axis and the distance from either focus to the ellipse's centre, eccentricity may be calculated. Mathematically, (1.2).

 

𝜀 = \(\frac{c_0}{a}\) = \(\frac{\sqrt{a^2-b^2}}{a}\) (1.2)

 

We can infer from (1.2) that eccentricity is inversely proportional to the length of the major axis and directly proportional to the distance between the centre and the focus. As eccentricity is a ratio between two lengths, it lacks a standard unit.

 

Circular Orbits

Because is equal to c₀, circular orbits have an eccentricity of 0. There is a fixedc0 distance from the centre at every point in the orbit. Circular orbits are automatically assumed to be confined orbits with a radius equal to any point on the circle and a constant orbital velocity. These orbits are just hypothesised in the case of planetary orbits.

 

Elliptical Orbits

An orbit with a range of eccentricities is called an elliptical orbit 0 < . The part of the orbit that is closest to the central body is known as periapsis, while the part that is farthest is known as apoapsis. The imbalance between the centrifugal force and the gravitational pull of the centre mass causes the planetary orbits to be elliptical. Moreover, this results in a variation in the orbital velocity across the elliptical orbit.

 

Parabolic Orbits:

When the eccentricity is equal to 1 of an orbit, it is known to be a parabolic orbit. This orbit can mostly be observed in comets as their velocity is equal to their escape velocity.

 

Hyperbolic Orbits:

Hyperbolic orbits are those that have eccentricities larger than one. These orbits usually avoid being pulled in by planets' gravitational fields. Objects that are prone to evading the gravitational attraction of planets frequently have these orbits. Hyperbolic Satellites frequently use orbits.

A planar curve defined by the conic section property that the sum of the lengths between any two points on the curve and the two fixed points is constant is called an ellipse. A plane intersecting a right-angled cone at an angle to its base creates an ellipse. The ellipse's equation in polar coordinates (2.1) and cartesian coordinates (2.0) is given by,

 

\(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=\) 1 (2.0)

\(\frac{a(1-e^2)}{r}=\) 1 + 𝑒𝑐𝑜𝑠θ (2.1)

Major Axis

The diameter of the Main Axis is the largest. The principal axis may shift vertically or horizontally.

 

Center

The Major axis's midway can be found at the centre of an ellipse. It is designated as c in figure (2).

 

Focus( Plural-Foci):

The focus or foci of each fixed point from which the distance is gauged to be constant ellipse. The midpoints between the ellipse's vertices' centres are known as the foci. They bear tags. as F₁ well F₂ as Figure (2).

 

Vertices

The Main Axis's endpoints are the ellipse's vertices.

 

Semi-Major Axis

The semi-major axis is the Major Axis' halved length. In the ellipse equation, the semi-major axis is indicated by (2.0) and (2.1)

 

Semi Minor Axis

The shorter radius of the ellipse's semi-minor axis is half its length. It is indicated in the graphic and the ellipse equation with the symbol.

 

Eccentricity

Eccentricity is a term used to describe how far an ellipse deviates from a circle, as was described in the essay's section on orbital categorization. The ratios between the distances between the centre and one of the foci and the semi-major axis must be larger than 0 but less than 1 on an ellipse.

Tycho Brahe, a collaborator of Kepler's, made astronomical advances by amassing empirical measurements with far better accuracy than had previously been possible7. Then, in 1609, Kepler published Astronomia Nova, which completely altered the field of celestial mechanics. Based on Tycho Brahe's findings, he developed principles that characterise the motion of planets in this.

 

Elliptical Orbit Law

"Each planet travels in an ellipse with the sun at one focus."

Kepler proposed the Elliptical Orbit law as an addition to the Copernican theory. Because ellipses were obviously less perfect than circles, Kepler thought that elliptical orbits were nothing more than an absurd ad hoc theory. 9 First law of Kepler was before Sir Issac Newton's publication of the Laws of Gravitation. Naturally, the agreement between the forecasts and the observations satisfied Kepler's first law. Nonetheless, this law has been repeatedly confirmed by alternative techniques even after Newton's Synthesis.

According to Newton's Second Law, the ellipse and orbit can be rewritten as follows:

 

𝐹 = 𝑚𝑎(3.0)

 

We also know from Newton’s Law of Gravitation that,

 

F = \(\frac{GMm}{r^2}\) (3.1)

 

From circular motion, we also know that,

 

 

ω = \(\frac{\theta}{t}\) and,

𝑎 = 𝑟ω² (3.2)

 

The acceleration 𝑟ω 2 will be the sum because the planet is travelling in a circular orbit and is still accelerating inward.

 

\(\frac{d^2r}{dt^2}- 𝑟ω^2=-\frac{GM}{r^2}\)

 

As the angular J momentum would be constant while the fluctuates ω, we can eliminate using J = mr².

 

\(\frac{d^2r}{dt^2} = - \frac{GM}{r}+r(\frac{J}{mr^2})^2\)

= \(-\frac{GM}{r^2}+\frac{j^2}{mr^3}\)

 

Integrating these equations,

𝐽 = 𝑚𝑟² ω = \(\frac{m}{u^2}\frac{d\theta}{dt}\) (3.3)

 

\(\frac{d}{dt}=\frac{ju^2}{m}\frac{d}{d\theta}\)

 

∴ \(\frac{dr}{dt}=\frac{d}{dt}(\frac1 u)=-\frac 1 u\frac{du}{dt}=-\frac Jm\frac{d}{d\theta}\) 

and

\(\frac{d^2r}{dt^2}=-\frac{J^2u^2}{m^2}\frac{d^2u}{d\theta^2}\)

 

Substituting,

\(\frac{d^2}{d\theta^2}+u=\frac{GMm^2}{j^2}\)

 

Solving,

 

𝑢 = \(\frac{1}r=\frac{GMm^2}{J}+\) AcosΘ

 

Here,

𝐴 = 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑜𝑓 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑖𝑜𝑛

As this is similar to the ellipse's standard equation, the relationship between it and the planet's orbit can be expressed by the equation,

\(\frac{a(1-e^2)}{r}=\) 1 + 𝑒𝑐𝑜𝑠θ (3.4)

 

Equal Area Law

“The radius vector from the sun to a planet sweeps out equal areas in equal time.”

 

𝑎𝑟𝑒𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = \(\frac{Δ𝐴}{Δt}=\frac{L}{2m}\) (3.5)

 

The conservation of angular momentum results in the equal-area law. Because the body experiences no torque from gravitational forces, its angular momentum ought to remain constant. As a result, this suggests that the orbit's time period and the body's surface area are connected.

 

Period Law:

"The semi-major axis of the an ellipse is related to the period of revolution T of a planet about the sun by where is constant for all planets." s𝑇² = 𝑘 𝑎³

 

With the discovery and solution of the planets' equation of motion in 1619, when the third law was published, there have been two centuries of mathematical and scientific advancement; the third planetary law, in contrast to the other two, was written about in a distinct book called Harmonices Mundi. The third law explains how the semi-major axis and the length of time it takes for a revolution to occur. Moreover, the orbital velocity formula is derived from this law.

 

F = \(\frac{GMm}{r^2}\) (3.6)

F = \(\frac{mv^2}{r}\) (3.7)

From (3.6) and (3.7)

 

v = \(\sqrt{\frac{GM}{r}}\) (3.8)

 

𝐴𝑙𝑠𝑜,

v = \(\frac{2\pi r}T\) (3.9)

 

𝑆𝑞𝑢𝑎𝑟𝑖𝑛𝑔 𝑣,

𝑣² = \(\frac{GM}r=\frac{4\pi^2r^2}{T^2}\)

\(\frac{GM}{r}=\frac{4\pi^2r^2}{T^2}\)

 

𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑖𝑛𝑔 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑤𝑖𝑡ℎ 𝑟 𝑎𝑛𝑑 𝑑𝑖𝑣𝑖𝑑𝑖𝑛𝑔 𝑤𝑖𝑡ℎ 4π²,

 

\(\frac{GM}{4\pi^2}\) = \(\frac{r^3}{T^2}\)

As \(\frac{GM}{4\pi^2}\) is a constant,

 

\(\frac{r^3}{T^2}=\) K

r³ = T²

𝑆𝑖𝑛𝑐𝑒 𝑟 = 𝑎

a³ = T² (4.0)

 

From this we can infer that the time period squared is equal to the cube of the semi-major axis.

With a mean eccentricity (𝜀) of 0.04, the known planets in the solar system have orbits that are almost round. Yet gas giant exoplanets exhibit a wide variety of anomalies. Moreover, the Orbital Eccentricity Formula can be used to deduce that Orbital Eccentricity is influenced by the separation between foci and the length of the primary axis (1.2). The orbital equation yields,

 

r = \(\frac{c}{1- 𝜀\ cos\theta}\) (4.1)

 

Rearranging the values,

 

 

𝑐 = \(\frac{J}{GMm^2}\) (4.2)

 

Orbital Velocity:

 

In contrast to a circular orbit, an elliptical orbit has a variable path about the center. As a result, The Orbital Velocity differs in the various regions of the ellipse. The body in orbit has constant mechanical energy because of the energy conservation law. The body's kinetic energy, however, tends to be highest in periapsis and lowest in apoapsis. Because the kinetic energy of a body is directly proportional to its velocity, the periapsis is also the region of the body where its orbital velocity is highest. It is also possible to discern the inverse relationship between orbital velocity and solar distance. The mean of the data is typically considered when determining the value of the orbital velocity.

 

The formula for the Orbital Velocity(3.9) can be derived from Kepler's third law.

 

𝑣 =  \(\dfrac{2 \pi a}{T}\)

 

𝑤ℎ𝑒𝑟𝑒,

 

𝑣 = 𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦

 

𝑇 = 𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑃𝑒𝑟𝑖𝑜𝑑

 

𝑎 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑒𝑚𝑖 𝑚𝑎𝑗𝑜𝑟 𝑎𝑥𝑖𝑠

 

Angular Momentum:

 

An object's momentum around an axis is measured by its angular momentum. The magnitude of the system's angular momentum, which is the vector sum of the stellar and planetary momenta, equals the magnitude of the sum. The general formula for the angular momentum is (4.3).

 

𝐽 = 𝑚𝑟² ω(4.3)

 

 

𝑊ℎ𝑒𝑟𝑒,

 

 

𝐽 = 𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚

 

The velocity and distance from the Sun both change as the planet move in an elliptical orbit, but the product of the velocity times the distance stays constant.

 

 

𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚(𝐽) = \(\dfrac{2 \pi m r^2}{p}\)

 

 

Angular Momentum is classified into two types:

 

Rotational Angular Momentum( ): The Rotational Angular Momentum is the momentum𝐽 𝑟𝑜𝑡 consumed in the body’s individual rotational motion.

 

𝐽𝑟𝑜𝑡 = 𝐼 × 𝑤

 

𝑊ℎ𝑒𝑟𝑒,

 

𝐼 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎

 

𝑤 = 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

 

The momentum consumedJorb throughout the body's orbital motion is known as the orbital angular momentum, or OAM. The conservation of energy has an impact on the orbital angular momentum. The planet is launched into orbit by the conservation of the body's original energy throughout the process of planet formation, from which it derives its orbital angular momentum.

 

𝐽_𝑜𝑟𝑏 = 𝑀𝑅² × 2 × \(\dfrac{\pi }{t_{orb}}\)

 

 

𝑊ℎ𝑒𝑟𝑒,

 

𝑀 = 𝑀𝑎𝑠𝑠 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒

 

 

𝑅 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑠𝑝ℎ𝑒𝑟𝑒

 

𝑇_𝑜𝑟𝑏 = 𝑂𝑟𝑏𝑖𝑡𝑎𝑙 𝑃𝑒𝑟𝑖𝑜𝑑 𝑜𝑓 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛, 𝑖𝑛 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

 

The total angular momentum is conserved as no net torque has been applied.

We must consider the origin of the orbital shape in order to assess the impact of the orbital velocity on the orbital eccentricity. When compared to the semi-major axis, the distance between the centre and focus increases an ellipse's eccentricity. The radius and semi-major axis of a circle are the same. The net forces are equal because the centrifugal and gravitational forces on the body are both constant. The orbit would only have one centre if the forces were distributed in a 1:1 ratio, and we would meet the requirements for a circular orbit. If, for example, a random kick velocity causes the planet's outward speed to be less or more than the gravitational pull, the orbital shape will stretch into an ellipse. This reasoning leads us to conclude that the orbital geometry of a body can be influenced by its orbital velocity. The orbital direction would also alter if the body lost a large amount of mass to the point where the centrifugal force was greater than the gravitational pull of the central body. The positions of perihelion would shift if the central body's mass decreased since a lower mass would result in a stronger gravitational attraction on the body. That would also result in a smaller semi-major axis, which would further reduce eccentricity as the orbit geometry moved closer to a circular shape. An alternative perspective would be to consider the energy contained within the orbit. Because of instability brought on by the variation in kinetic energy across the orbit, the center-focus distance and semi-major axis' relative distances will eventually increase.

 

It can be inferred from the conservation of angular momentum that the orbital velocity is either minimum at the aphelion or maximum at the perihelion. The point where the direction of the outward velocity and the gravitational force coincide would be where the velocity would be at its highest. Considering a situation in which the orbital velocity varies while the other elements stay the same,

 

Let us take cases,

 

Case I: The orbit with orbital velocity is 𝑣

v = \(\frac{2\pi a}{T}\)

 

Case II: The orbit with orbital velocity 2𝑣 = 𝑣₀

 

𝑣 ₀  = \(\frac{4\pi a}T\)

 

Recalling some previously used equations,

 

𝑣 = \(\frac{2\pi a}T\) (3.9)

 

 

𝜀 = \(\frac{c_0}{a}\) (1.2)

(3.9) can be rewritten as,

\(\frac{vT}{2\pi}=\)  a (4.4)

For Case I:

v = \(\frac{2\pi a}T\) 

a = \(\frac{vT}{2 \pi}\) (4.5)

 

Replacing (4.5) at the eccentricity equation,

\(𝜀 = \frac{c_0 \times2\pi}{vT}\) (4.6)

 

For Case II:

 

𝑣 ₀ = \(\frac{4\pi a}{T}\)

𝑎₀ = \(\frac{v_0T}{4\pi}\)

𝜀 = \(\frac{c_0}{a}\) (1.2)

 

ε ₀  = \(\frac{c_04\pi}{vt}\) (4.7)

 

Comparing Eccentricities (4.6) and (4.7),

\(\frac{ε_ 0}{ε}=\frac{c_0\times4\pi}{vT}\times\frac{vT}{c_0\times2\pi}\)

\(\frac{ε\ _0}{ε}=\frac{2}1\)

ε ₀ = 2ε

so

 

When the 𝑣₀ = 2𝑣 the ε ₀ = 2ε

 

The outcome implies that the highest orbital velocities are found in the most eccentric orbits. Table (1)'s planetary data show that there are some exceptions to this correlation. For instance, table (1)'s solar system information reveals that Venus' orbit is extremely close to It's in a circle, but it moves at a fast speed. Several variables that affect orbital eccentricity may have contributed to this. The analysis we came to as a conclusion with looks at a single planet under the gravitational pull of the main object. The solar system is a closed system made up of numerous bodies, each of which has a distinct gravitational field. As more forces and bodies are involved in this situation, the outcome gets significantly more convoluted. in the formula. If we take into account the body's position in relation to the other bodies as gravitation increases as the distance between the masses grows, further problems would result.decreases.

In conclusion, the Angular Momentum rather conducts the change in velocities and velocities might effect the orbital geometry in order to respond to the study question, "How do Angular Momentum and Orbital Velocity affect the Eccentricity of Planetary Orbits?" According to Kepler's Second Law, the system's rotational momentum is always conserved. This suggests that in order to make up for the reduced velocity, the orbital velocity is varied in each section of the elliptical orbit. There is a force imbalance as an orbit becomes elliptical. For instance, the eccentricity of the orbit will alter if the gravitational force of the central mass is reduced or the orbiting body's velocity is increased by a kick velocity. The Orbit would become elliptical if it had been circular in the prior scenario. Further analysis led to the conclusion that if the body's velocity had increased, the eccentricity would also have grown because the centrifugal force would have increased at the same time. Additionally, it suggests that planets with the fastest orbital velocity would be more likely to have orbits that are more eccentric. Exceptions were also included because there are other things that can impact orbital geometry. If Kepler's Third Law is used as a lens, this result can also be seen from a different angle. Also, since the system's angular momentum would be conserved, the orbital eccentricity wouldn't be the main factor affected. For instance, if the beginning energy was identical, the angular momentum within a circular orbit might be the same as for an elliptical orbit. Also incorporating mass into the equation is angular momentum.

Just so much can be covered in a 4000-word essay. Two of these factors were the focus of this essay, however there are still other aspects that might have an impact on the result. There are still other factors that have not been considered. These factors have been briefly mentioned in this section so they can be investigated if more research on this subject is done. A wide range of potential faults are also covered in this section along with explanations.

 

• Each planet has a distinct separation from its primary mass, which alters the gravitational pull.
• The essay discusses planets generally, although each type of planet also has its own categories, such as Gas Giants. The mass of gas giants is typically higher.
• As angular momentum depends on mass, changing the mass may also alter the eccentricity. The torque, which influences the overall angular momentum, may be impacted by a quick shift in mass. The conservation of angular momentum is broken when an external tension is applied.
• The equation is only relevant as long as a third body isn’t introduced to it. Famously known as the three-body problem, the predictions would be off if a third body is introduced.
• The orbital equation assumes that the planet's motion is periodic, but general relativity has taught us that planets do not move in a periodic way.
• If the data source had uncertainties for the data in the table, the estimations could be more accurate.

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