Kepler orbit

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https://en.wikipedia.org/wiki/Kepler_orbit

An elliptic Kepler orbit with an eccentricity of 0.7, a parabolic Kepler orbit and a hyperbolic Kepler orbit with an eccentricity of 1.3. The distance to the focal point is a function of the polar angle relative to the horizontal line as given by the equation (13)

In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

In most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of similar mass can be described as Kepler orbits around their common center of mass, their barycenter.

Introduction

From ancient times until the 16th and 17th centuries, the motions of the planets were believed to follow perfectly circular geocentric paths as taught by the ancient Greek philosophers Aristotle and Ptolemy. Variations in the motions of the planets were explained by smaller circular paths overlaid on the larger path (see epicycle). As measurements of the planets became increasingly accurate, revisions to the theory were proposed. In 1543, Nicolaus Copernicus published a heliocentric model of the Solar System, although he still believed that the planets traveled in perfectly circular paths centered on the Sun.

Development of the laws

In 1601, Johannes Kepler acquired the extensive, meticulous observations of the planets made by Tycho Brahe. Kepler would spend the next five years trying to fit the observations of the planet Mars to various curves. In 1609, Kepler published the first two of his three laws of planetary motion. The first law states:

The orbit of every planet is an ellipse with the sun at a focus.

More generally, the path of an object undergoing Keplerian motion may also follow a parabola or a hyperbola, which, along with ellipses, belong to a group of curves known as conic sections. Mathematically, the distance between a central body and an orbiting body can be expressed as:

$${\displaystyle r(\theta )=\frac{a(1-e^2)}{1+e\cos (\theta )}}$$

where:

• ${\displaystyle r}$ is the distance
• ${\displaystyle a}$ is the semi-major axis, which defines the size of the orbit
• ${\displaystyle e}$ is the eccentricity, which defines the shape of the orbit
• ${\displaystyle \theta }$ is the true anomaly, which is the angle between the current position of the orbiting object and the location in the orbit at which it is closest to the central body (called the periapsis).

Alternately, the equation can be expressed as:

$${\displaystyle r(\theta )=\frac{p}{1+e\cos (\theta )}}$$

Where ${\displaystyle p}$ is called the semi-latus rectum of the curve. This form of the equation is particularly useful when dealing with parabolic trajectories, for which the semi-major axis is infinite.

Despite developing these laws from observations, Kepler was never able to develop a theory to explain these motions.

Isaac Newton

Between 1665 and 1666, Isaac Newton developed several concepts related to motion, gravitation and differential calculus. However, these concepts were not published until 1687 in the Principia, in which he outlined his laws of motion and his law of universal gravitation. His second of his three laws of motion states:

The acceleration of a body is parallel and directly proportional to the net force acting on the body, is in the direction of the net force, and is inversely proportional to the mass of the body:

$${\displaystyle \mathbf{F}=m\mathbf{a}=m\frac{d^2\mathbf{r}}{d{t}^2}}$$

Where:

• ${\displaystyle \mathbf{F}}$ is the force vector
• ${\displaystyle m}$ is the mass of the body on which the force is acting
• ${\displaystyle \mathbf{a}}$ is the acceleration vector, the second time derivative of the position vector ${\displaystyle \mathbf{r}}$

Strictly speaking, this form of the equation only applies to an object of constant mass, which holds true based on the simplifying assumptions made below.

The mechanisms of Newton's law of universal gravitation; a point mass m₁ attracts another point mass m₂ by a force F₂ which is proportional to the product of the two masses and inversely proportional to the square of the distance (r) between them. Regardless of masses or distance, the magnitudes of |F₁| and |F₂| will always be equal. G is the gravitational constant.

Newton's law of gravitation states:

Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses:

$${\displaystyle F=G\frac{m_1{m}_2}{r^2}}$$

where:

• ${\displaystyle F}$ is the magnitude of the gravitational force between the two point masses
• ${\displaystyle G}$ is the gravitational constant
• ${\displaystyle m_1}$ is the mass of the first point mass
• ${\displaystyle m_2}$ is the mass of the second point mass
• ${\displaystyle r}$ is the distance between the two point masses

From the laws of motion and the law of universal gravitation, Newton was able to derive Kepler's laws, which are specific to orbital motion in astronomy. Since Kepler's laws were well-supported by observation data, this consistency provided strong support of the validity of Newton's generalized theory, and unified celestial and ordinary mechanics. These laws of motion formed the basis of modern celestial mechanics until Albert Einstein introduced the concepts of special and general relativity in the early 20th century. For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy and is used extensively in astronomy and astrodynamics.

Simplified two body problem

See also: Orbit § Newtonian analysis of orbital motion

To solve for the motion of an object in a two body system, two simplifying assumptions can be made:

1. The bodies are spherically symmetric and can be treated as point masses.
2. There are no external or internal forces acting upon the bodies other than their mutual gravitation.

The shapes of large celestial bodies are close to spheres. By symmetry, the net gravitational force attracting a mass point towards a homogeneous sphere must be directed towards its centre. The shell theorem (also proven by Isaac Newton) states that the magnitude of this force is the same as if all mass was concentrated in the middle of the sphere, even if the density of the sphere varies with depth (as it does for most celestial bodies). From this immediately follows that the attraction between two homogeneous spheres is as if both had its mass concentrated to its center.

Smaller objects, like asteroids or spacecraft often have a shape strongly deviating from a sphere. But the gravitational forces produced by these irregularities are generally small compared to the gravity of the central body. The difference between an irregular shape and a perfect sphere also diminishes with distances, and most orbital distances are very large when compared with the diameter of a small orbiting body. Thus for some applications, shape irregularity can be neglected without significant impact on accuracy. This effect is quite noticeable for artificial Earth satellites, especially those in low orbits.

Planets rotate at varying rates and thus may take a slightly oblate shape because of the centrifugal force. With such an oblate shape, the gravitational attraction will deviate somewhat from that of a homogeneous sphere. At larger distances the effect of this oblateness becomes negligible. Planetary motions in the Solar System can be computed with sufficient precision if they are treated as point masses.

Two point mass objects with masses ${\displaystyle m_1}$ and ${\displaystyle m_2}$ and position vectors ${\displaystyle {\mathbf{r}}_1}$ and ${\displaystyle {\mathbf{r}}_2}$ relative to some inertial reference frame experience gravitational forces:

$${\displaystyle m_1{\ddot{\mathbf{r}}}_1=\frac{-G{m}_1{m}_2}{r^2}\hat{\mathbf{r}}}$$

$${\displaystyle m_2{\ddot{\mathbf{r}}}_2=\frac{G{m}_1{m}_2}{r^2}\hat{\mathbf{r}}}$$

where ${\displaystyle \mathbf{r}}$ is the relative position vector of mass 1 with respect to mass 2, expressed as:

$${\displaystyle \mathbf{r}={\mathbf{r}}_1-{\mathbf{r}}_2}$$

and ${\displaystyle \hat{\mathbf{r}}}$ is the unit vector in that direction and ${\displaystyle r}$ is the length of that vector.

Dividing by their respective masses and subtracting the second equation from the first yields the equation of motion for the acceleration of the first object with respect to the second:

${\displaystyle \ddot{\mathbf{r}}=-\frac{\alpha }{r^2}\hat{\mathbf{r}}}$

(1)

where ${\displaystyle \alpha }$ is the gravitational parameter and is equal to

$${\displaystyle \alpha =G(m_1+m_2)}$$

In many applications, a third simplifying assumption can be made:

3. When compared to the central body, the mass of the orbiting body is insignificant. Mathematically, m₁ >> m₂, so α = G (m₁ + m₂) ≈ Gm₁. Such standard gravitational parameters, often denoted as ${\displaystyle \mu =GM}$, are widely available for Sun, major planets and Moon, which have much larger masses ${\displaystyle M}$ than their orbiting satellites.

This assumption is not necessary to solve the simplified two body problem, but it simplifies calculations, particularly with Earth-orbiting satellites and planets orbiting the Sun. Even Jupiter's mass is less than the Sun's by a factor of 1047, which would constitute an error of 0.096% in the value of α. Notable exceptions include the Earth-Moon system (mass ratio of 81.3), the Pluto-Charon system (mass ratio of 8.9) and binary star systems.

Under these assumptions the differential equation for the two body case can be completely solved mathematically and the resulting orbit which follows Kepler's laws of planetary motion is called a "Kepler orbit". The orbits of all planets are to high accuracy Kepler orbits around the Sun. The small deviations are due to the much weaker gravitational attractions between the planets, and in the case of Mercury, due to general relativity. The orbits of the artificial satellites around the Earth are, with a fair approximation, Kepler orbits with small perturbations due to the gravitational attraction of the Sun, the Moon and the oblateness of the Earth. In high accuracy applications for which the equation of motion must be integrated numerically with all gravitational and non-gravitational forces (such as solar radiation pressure and atmospheric drag) being taken into account, the Kepler orbit concepts are of paramount importance and heavily used.

Keplerian elements

Keplerian orbital elements.

Main article: Keplerian elements

Any Keplerian trajectory can be defined by six parameters. The motion of an object moving in three-dimensional space is characterized by a position vector and a velocity vector. Each vector has three components, so the total number of values needed to define a trajectory through space is six. An orbit is generally defined by six elements (known as Keplerian elements) that can be computed from position and velocity, three of which have already been discussed. These elements are convenient in that of the six, five are unchanging for an unperturbed orbit (a stark contrast to two constantly changing vectors). The future location of an object within its orbit can be predicted and its new position and velocity can be easily obtained from the orbital elements.

Two define the size and shape of the trajectory:

• Semimajor axis (${\displaystyle a}$)
• Eccentricity (${\displaystyle e}$)

Three define the orientation of the orbital plane:

• Inclination (${\displaystyle i}$) defines the angle between the orbital plane and the reference plane.
• Longitude of the ascending node (${\displaystyle \mathrm{\Omega }}$) defines the angle between the reference direction and the upward crossing of the orbit on the reference plane (the ascending node).
• Argument of periapsis (${\displaystyle \omega }$) defines the angle between the ascending node and the periapsis.

And finally:

• True anomaly (${\displaystyle \nu }$) defines the position of the orbiting body along the trajectory, measured from periapsis. Several alternate values can be used instead of true anomaly, the most common being ${\displaystyle M}$ the mean anomaly and ${\displaystyle T}$, the time since periapsis.

Because ${\displaystyle i}$, ${\displaystyle \mathrm{\Omega }}$ and ${\displaystyle \omega }$ are simply angular measurements defining the orientation of the trajectory in the reference frame, they are not strictly necessary when discussing the motion of the object within the orbital plane. They have been mentioned here for completeness, but are not required for the proofs below.

Mathematical solution of the differential equation (1) above

For movement under any central force, i.e. a force parallel to r, the specific relative angular momentum ${\displaystyle \mathbf{H}=\mathbf{r}\times \dot{\mathbf{r}}}$ stays constant:

$${\displaystyle \dot{\mathbf{H}}=\frac{d}{dt}\left(\mathbf{r}\times \dot{\mathbf{r}}\right)=\dot{\mathbf{r}}\times \dot{\mathbf{r}}+\mathbf{r}\times \ddot{\mathbf{r}}=\mathbf{0}+\mathbf{0}=\mathbf{0}}$$

Since the cross product of the position vector and its velocity stays constant, they must lie in the same plane, orthogonal to ${\displaystyle \mathbf{H}}$. This implies the vector function is a plane curve.

Because the equation has symmetry around its origin, it is easier to solve in polar coordinates. However, it is important to note that equation (1) refers to linear acceleration ${\displaystyle \left(\ddot{\mathbf{r}}\right),}$ as opposed to angular ${\displaystyle \left(\ddot{\theta }\right)}$ or radial ${\displaystyle \left(\ddot{r}\right)}$ acceleration. Therefore, one must be cautious when transforming the equation. Introducing a cartesian coordinate system ${\displaystyle (\hat{\mathbf{x}},\hat{\mathbf{y}})}$ and polar unit vectors ${\displaystyle (\hat{\mathbf{r}},\hat{\mathbf{q}})}$ in the plane orthogonal to ${\displaystyle \mathbf{H}}$:

$${\displaystyle \begin{array}{rl}\hat{\mathbf{r}}& =\cos \theta \hat{\mathbf{x}}+\sin \theta \hat{\mathbf{y}}\\ \hat{\mathbf{q}}& =-\sin \theta \hat{\mathbf{x}}+\cos \theta \hat{\mathbf{y}}\end{array}}$$

We can now rewrite the vector function ${\displaystyle \mathbf{r}}$ and its derivatives as:

$${\displaystyle \begin{array}{rl}\mathbf{r}& =r\left(\cos \theta \hat{\mathbf{x}}+\sin \theta \hat{\mathbf{y}}\right)=r\hat{\mathbf{r}}\\ \dot{\mathbf{r}}& =\dot{r}\hat{\mathbf{r}}+r\dot{\theta }\hat{\mathbf{q}}\\ \ddot{\mathbf{r}}& =\left(\ddot{r}-r{\dot{\theta }}^2\right)\hat{\mathbf{r}}+\left(r\ddot{\theta }+2\dot{r}\dot{\theta }\right)\hat{\mathbf{q}}\end{array}}$$

(see "Vector calculus"). Substituting these into (1), we find:

$${\displaystyle \left(\ddot{r}-r{\dot{\theta }}^2\right)\hat{\mathbf{r}}+\left(r\ddot{\theta }+2\dot{r}\dot{\theta }\right)\hat{\mathbf{q}}=\left(-\frac{\alpha }{r^2}\right)\hat{\mathbf{r}}+(0)\hat{\mathbf{q}}}$$

This gives the ordinary differential equation in the two variables ${\displaystyle r}$ and ${\displaystyle \theta }$:

${\displaystyle \ddot{r}-r{\dot{\theta }}^2=-\frac{\alpha }{r^2}}$

(2)

In order to solve this equation, all time derivatives must be eliminated. This brings:

$${\displaystyle H=|\mathbf{r}\times \dot{\mathbf{r}}|=\left|\left(\begin{array}{c}r\cos (\theta )\\ r\sin (\theta )\\ 0\end{array}\right)\times \left(\begin{array}{c}\dot{r}\cos (\theta )-r\sin (\theta )\dot{\theta }\\ \dot{r}\sin (\theta )+r\cos (\theta )\dot{\theta }\\ 0\end{array}\right)\right|=\left|\left(\begin{array}{c}0\\ 0\\ r^2\dot{\theta }\end{array}\right)\right|=r^2\dot{\theta }}$$

${\displaystyle \dot{\theta }=\frac{H}{r^2}}$

(3)

Taking the time derivative of (3) gets

${\displaystyle \ddot{\theta }=-\frac{2\cdot H\cdot \dot{r}}{r^3}}$

(4)

Equations (3) and (4) allow us to eliminate the time derivatives of ${\displaystyle \theta }$. In order to eliminate the time derivatives of ${\displaystyle r}$, the chain rule is used to find appropriate substitutions:

${\displaystyle \dot{r}=\frac{dr}{d\theta }\cdot \dot{\theta }}$

(5)

${\displaystyle \ddot{r}=\frac{d^{2}r}{d{\theta }^2}\cdot {\dot{\theta }}^2+\frac{dr}{d\theta }\cdot \ddot{\theta }}$

(6)

Using these four substitutions, all time derivatives in (2) can be eliminated, yielding an ordinary differential equation for ${\displaystyle r}$ as function of ${\displaystyle \theta .}$

$${\displaystyle \ddot{r}-r{\dot{\theta }}^2=-\frac{\alpha }{r^2}}$$

$${\displaystyle \frac{d^{2}r}{d{\theta }^2}\cdot {\dot{\theta }}^2+\frac{dr}{d\theta }\cdot \ddot{\theta }-r{\dot{\theta }}^2=-\frac{\alpha }{r^2}}$$

$${\displaystyle \frac{d^{2}r}{d{\theta }^2}\cdot {\left(\frac{H}{r^2}\right)}^2+\frac{dr}{d\theta }\cdot \left(-\frac{2\cdot H\cdot \dot{r}}{r^3}\right)-r{\left(\frac{H}{r^2}\right)}^2=-\frac{\alpha }{r^2}}$$

${\displaystyle \frac{H^2}{r^4}\cdot \left(\frac{d^{2}r}{d{\theta }^2}-2\cdot \frac{{\left(\frac{dr}{d\theta }\right)}^2}{r}-r\right)=-\frac{\alpha }{r^2}}$

(7)

The differential equation (7) can be solved analytically by the variable substitution

${\displaystyle r=\frac{1}{s}}$

(8)

Using the chain rule for differentiation gets:

${\displaystyle \frac{dr}{d\theta }=-\frac{1}{s^2}\cdot \frac{ds}{d\theta }}$

(9)

${\displaystyle \frac{d^{2}r}{d{\theta }^2}=\frac{2}{s^3}\cdot {\left(\frac{ds}{d\theta }\right)}^2-\frac{1}{s^2}\cdot \frac{d^{2}s}{d{\theta }^2}}$

(10)

Using the expressions (10) and (9) for ${\displaystyle \frac{d^{2}r}{d{\theta }^2}}$ and ${\displaystyle \frac{dr}{d\theta }}$ gets

${\displaystyle H^2\cdot \left(\frac{d^{2}s}{d{\theta }^2}+s\right)=\alpha }$

(11)

with the general solution

${\displaystyle s=\frac{\alpha }{H^2}\cdot \left(1+e\cdot \cos (\theta -{\theta }_0)\right)}$

(12)

where e and ${\displaystyle {\theta }_0}$ are constants of integration depending on the initial values for s and ${\displaystyle \frac{ds}{d\theta }.}$

Instead of using the constant of integration ${\displaystyle {\theta }_0}$ explicitly one introduces the convention that the unit vectors ${\displaystyle \hat{x},\hat{y}}$ defining the coordinate system in the orbital plane are selected such that ${\displaystyle {\theta }_0}$ takes the value zero and e is positive. This then means that ${\displaystyle \theta }$ is zero at the point where ${\displaystyle s}$ is maximal and therefore ${\displaystyle r=\frac{1}{s}}$ is minimal. Defining the parameter p as ${\displaystyle \frac{H^2}{\alpha }}$ one has that

$${\displaystyle r=\frac{1}{s}=\frac{p}{1+e\cdot \cos \theta }}$$

Alternate derivation

Another way to solve this equation without the use of polar differential equations is as follows:

Define a unit vector ${\displaystyle \mathbf{u}}$, ${\displaystyle \mathbf{u}=\frac{\mathbf{r}}{r}}$, such that ${\displaystyle \mathbf{r}=r\mathbf{u}}$ and ${\displaystyle \ddot{\mathbf{r}}=-\frac{\alpha }{r^2}\mathbf{u}}$. It follows that

$${\displaystyle \mathbf{H}=\mathbf{r}\times \dot{\mathbf{r}}=r\mathbf{u}\times \frac{d}{dt}(r\mathbf{u})=r\mathbf{u}\times (r\dot{\mathbf{u}}+\dot{r}\mathbf{u})=r^2(\mathbf{u}\times \dot{\mathbf{u}})+r\dot{r}(\mathbf{u}\times \mathbf{u})=r^2\mathbf{u}\times \dot{\mathbf{u}}}$$

Now consider

$${\displaystyle \ddot{\mathbf{r}}\times \mathbf{H}=-\frac{\alpha }{r^2}\mathbf{u}\times (r^2\mathbf{u}\times \dot{\mathbf{u}})=-\alpha \mathbf{u}\times (\mathbf{u}\times \dot{\mathbf{u}})=-\alpha [(\mathbf{u}\cdot \dot{\mathbf{u}})\mathbf{u}-(\mathbf{u}\cdot \mathbf{u})\dot{\mathbf{u}}]}$$

(see Vector triple product). Notice that

$${\displaystyle \mathbf{u}\cdot \mathbf{u}=|\mathbf{u}{|}^2=1}$$

$${\displaystyle \mathbf{u}\cdot \dot{\mathbf{u}}=\frac{1}{2}(\mathbf{u}\cdot \dot{\mathbf{u}}+\dot{\mathbf{u}}\cdot \mathbf{u})=\frac{1}{2}\frac{d}{dt}(\mathbf{u}\cdot \mathbf{u})=0}$$

Substituting these values into the previous equation gives:

$${\displaystyle \ddot{\mathbf{r}}\times \mathbf{H}=\alpha \dot{\mathbf{u}}}$$

Integrating both sides:

$${\displaystyle \dot{\mathbf{r}}\times \mathbf{H}=\alpha \mathbf{u}+\mathbf{c}}$$

where c is a constant vector. Dotting this with r yields an interesting result:

$${\displaystyle \mathbf{r}\cdot (\dot{\mathbf{r}}\times \mathbf{H})=\mathbf{r}\cdot (\alpha \mathbf{u}+\mathbf{c})=\alpha \mathbf{r}\cdot \mathbf{u}+\mathbf{r}\cdot \mathbf{c}=\alpha r(\mathbf{u}\cdot \mathbf{u})+rc\cos (\theta )=r(\alpha +c\cos (\theta ))}$$

where ${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbf{r}}$ and ${\displaystyle \mathbf{c}}$. Solving for r :

$${\displaystyle r=\frac{\mathbf{r}\cdot (\dot{\mathbf{r}}\times \mathbf{H})}{\alpha +c\cos (\theta )}=\frac{(\mathbf{r}\times \dot{\mathbf{r}})\cdot \mathbf{H}}{\alpha +c\cos (\theta )}=\frac{|\mathbf{H}{|}^2}{\alpha +c\cos (\theta )}=\frac{|\mathbf{H}{|}^2/\alpha }{1+(c/\alpha )\cos (\theta )}.}$$

Notice that ${\displaystyle (r,\theta )}$ are effectively the polar coordinates of the vector function. Making the substitutions ${\displaystyle p=\frac{|\mathbf{H}{|}^2}{\alpha }}$ and ${\displaystyle e=\frac{c}{\alpha }}$, we again arrive at the equation

${\displaystyle r=\frac{p}{1+e\cdot \cos \theta }}$

(13)

This is the equation in polar coordinates for a conic section with origin in a focal point. The argument ${\displaystyle \theta }$ is called "true anomaly".

Eccentricity Vector

Notice also that, since ${\displaystyle \theta }$ is the angle between the position vector ${\displaystyle \mathbf{r}}$ and the integration constant ${\displaystyle \mathbf{c}}$, the vector ${\displaystyle \mathbf{c}}$ must be pointing in the direction of the periapsis of the orbit. We can then define the eccentricity vector associated with the orbit as:

$${\displaystyle \mathbf{e}\triangleq \frac{\mathbf{c}}{\alpha }=\frac{\dot{\mathbf{r}}\times \mathbf{H}}{\alpha }-\mathbf{u}=\frac{\mathbf{v}\times \mathbf{H}}{\alpha }-\frac{\mathbf{r}}{r}=\frac{\mathbf{v}\times (\mathbf{r}\times \mathbf{v})}{\alpha }-\frac{\mathbf{r}}{r}}$$

where ${\displaystyle \mathbf{H}=\mathbf{r}\times \dot{\mathbf{r}}=\mathbf{r}\times \mathbf{v}}$ is the constant angular momentum vector of the orbit, and ${\displaystyle \mathbf{v}}$ is the velocity vector associated with the position vector ${\displaystyle \mathbf{r}}$.

Obviously, the eccentricity vector, having the same direction as the integration constant ${\displaystyle \mathbf{c}}$, also points to the direction of the periapsis of the orbit, and it has the magnitude of orbital eccentricity. This makes it very useful in orbit determination (OD) for the orbital elements of an orbit when a state vector [${\displaystyle \mathbf{r},\dot{\mathbf{r}}}$] or [${\displaystyle \mathbf{r},\mathbf{v}}$] is known.

Properties of trajectory equation

For ${\displaystyle e=0}$ this is a circle with radius p.

For ${\displaystyle 0<e<1,}$ this is an ellipse with

${\displaystyle a=\frac{p}{1-e^2}}$

(14)

${\displaystyle b=\frac{p}{\sqrt{1-e^2}}=a\cdot \sqrt{1-e^2}}$

(15)

For ${\displaystyle e=1}$ this is a parabola with focal length ${\displaystyle \frac{p}{2}}$

For ${\displaystyle e>1}$ this is a hyperbola with

${\displaystyle a=\frac{p}{e^2-1}}$

(16)

${\displaystyle b=\frac{p}{\sqrt{e^2-1}}=a\cdot \sqrt{e^2-1}}$

(17)

The following image illustrates a circle (grey), an ellipse (red), a parabola (green) and a hyperbola (blue)

A diagram of the various forms of the Kepler Orbit and their eccentricities. Blue is a hyperbolic trajectory (e > 1). Green is a parabolic trajectory (e = 1). Red is an elliptical orbit (0 < e < 1). Grey is a circular orbit (e = 0).

The point on the horizontal line going out to the right from the focal point is the point with ${\displaystyle \theta =0}$ for which the distance to the focus takes the minimal value ${\displaystyle \frac{p}{1+e},}$ the pericentre. For the ellipse there is also an apocentre for which the distance to the focus takes the maximal value ${\displaystyle \frac{p}{1-e}.}$ For the hyperbola the range for ${\displaystyle \theta }$ is

$${\displaystyle -{\cos }^{-1}\left(-\frac{1}{e}\right)<\theta <{\cos }^{-1}\left(-\frac{1}{e}\right)}$$

and for a parabola the range is

$${\displaystyle -\pi <\theta <\pi }$$

Using the chain rule for differentiation (5), the equation (2) and the definition of p as ${\displaystyle \frac{H^2}{\alpha }}$ one gets that the radial velocity component is

${\displaystyle V_r=\dot{r}=\frac{H}{p}e\sin \theta =\sqrt{\frac{\alpha }{p}}e\sin \theta }$

(18)

and that the tangential component (velocity component perpendicular to ${\displaystyle V_r}$) is

${\displaystyle V_t=r\cdot \dot{\theta }=\frac{H}{r}=\sqrt{\frac{\alpha }{p}}\cdot (1+e\cdot \cos \theta )}$

(19)

The connection between the polar argument ${\displaystyle \theta }$ and time t is slightly different for elliptic and hyperbolic orbits.

For an elliptic orbit one switches to the "eccentric anomaly" E for which

${\displaystyle x=a\cdot \cos E}$

(20)

${\displaystyle y=b\cdot \sin E}$

(21)

and consequently

${\displaystyle \dot{x}=-a\cdot \sin E\cdot \dot{E}}$

(22)

${\displaystyle \dot{y}=b\cdot \cos E\cdot \dot{E}}$

(23)

and the angular momentum H is

${\displaystyle H=x\cdot \dot{y}-y\cdot \dot{x}=a\cdot b\cdot (1-e\cdot \cos E)\cdot \dot{E}}$

(24)

Integrating with respect to time t gives

${\displaystyle H\cdot t=a\cdot b\cdot (E-e\cdot \sin E)}$

(25)

under the assumption that time ${\displaystyle t=0}$ is selected such that the integration constant is zero.

As by definition of p one has

${\displaystyle H=\sqrt{\alpha \cdot p}}$

(26)

this can be written

${\displaystyle t=a\cdot \sqrt{\frac{a}{\alpha }}(E-e\cdot \sin E)}$

(27)

For a hyperbolic orbit one uses the hyperbolic functions for the parameterisation

${\displaystyle x=a\cdot (e-\cosh E)}$

(28)

${\displaystyle y=b\cdot \sinh E}$

(29)

for which one has

${\displaystyle \dot{x}=-a\cdot \sinh E\cdot \dot{E}}$

(30)

${\displaystyle \dot{y}=b\cdot \cosh E\cdot \dot{E}}$

(31)

and the angular momentum H is

${\displaystyle H=x\cdot \dot{y}-y\cdot \dot{x}=a\cdot b\cdot (e\cdot \cosh E-1)\cdot \dot{E}}$

(32)

Integrating with respect to time t gets

${\displaystyle H\cdot t=a\cdot b\cdot (e\cdot \sinh E-E)}$

(33)

i.e.

${\displaystyle t=a\cdot \sqrt{\frac{a}{\alpha }}(e\cdot \sinh E-E)}$

(34)

To find what time t that corresponds to a certain true anomaly ${\displaystyle \theta }$ one computes corresponding parameter E connected to time with relation (27) for an elliptic and with relation (34) for a hyperbolic orbit.

Note that the relations (27) and (34) define a mapping between the ranges

$${\displaystyle \left[-\mathrm{\infty }<t<\mathrm{\infty }\right]⟷\left[-\mathrm{\infty }<E<\mathrm{\infty }\right]}$$

Some additional formulae

For an elliptic orbit one gets from (20) and (21) that

${\displaystyle r=a\cdot (1-e\cos E)}$

(35)

and therefore that

${\displaystyle \cos \theta =\frac{x}{r}=\frac{\cos E-e}{1-e\cos E}}$

(36)

From (36) then follows that

$${\displaystyle {\tan }^2\frac{\theta }{2}=\frac{1-\cos \theta }{1+\cos \theta }=\frac{1-\frac{\cos E-e}{1-e\cos E}}{1+\frac{\cos E-e}{1-e\cos E}}=\frac{1-e\cos E-\cos E+e}{1-e\cos E+\cos E-e}=\frac{1+e}{1-e}\cdot \frac{1-\cos E}{1+\cos E}=\frac{1+e}{1-e}\cdot {\tan }^2\frac{E}{2}}$$

From the geometrical construction defining the eccentric anomaly it is clear that the vectors ${\displaystyle (\cos E,\sin E)}$ and ${\displaystyle (\cos \theta ,\sin \theta )}$ are on the same side of the x-axis. From this then follows that the vectors ${\displaystyle \left(\cos \frac{E}{2},\sin \frac{E}{2}\right)}$ and ${\displaystyle \left(\cos \frac{\theta }{2},\sin \frac{\theta }{2}\right)}$ are in the same quadrant. One therefore has that

${\displaystyle \tan \frac{\theta }{2}=\sqrt{\frac{1+e}{1-e}}\cdot \tan \frac{E}{2}}$

(37)

and that

${\displaystyle \theta =2\cdot \arg \left(\sqrt{1-e}\cdot \cos \frac{E}{2},\sqrt{1+e}\cdot \sin \frac{E}{2}\right)+n\cdot 2\pi }$

(38)

${\displaystyle E=2\cdot \arg \left(\sqrt{1+e}\cdot \cos \frac{\theta }{2},\sqrt{1-e}\cdot \sin \frac{\theta }{2}\right)+n\cdot 2\pi }$

(39)

where "${\displaystyle \arg (x,y)}$" is the polar argument of the vector ${\displaystyle (x,y)}$ and n is selected such that ${\displaystyle |E-\theta |<\pi }$

For the numerical computation of ${\displaystyle \arg (x,y)}$ the standard function ATAN2(y,x) (or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN can be used.

Note that this is a mapping between the ranges

$${\displaystyle \left[-\mathrm{\infty }<\theta <\mathrm{\infty }\right]⟷\left[-\mathrm{\infty }<E<\mathrm{\infty }\right]}$$

For a hyperbolic orbit one gets from (28) and (29) that

${\displaystyle r=a\cdot (e\cdot \cosh E-1)}$

(40)

and therefore that

${\displaystyle \cos \theta =\frac{x}{r}=\frac{e-\cosh E}{e\cdot \cosh E-1}}$

(41)

As

$${\displaystyle {\tan }^2\frac{\theta }{2}=\frac{1-\cos \theta }{1+\cos \theta }=\frac{1-\frac{e-\cosh E}{e\cdot \cosh E-1}}{1+\frac{e-\cosh E}{e\cdot \cosh E-1}}=\frac{e\cdot \cosh E-e+\cosh E}{e\cdot \cosh E+e-\cosh E}=\frac{e+1}{e-1}\cdot \frac{\cosh E-1}{\cosh E+1}=\frac{e+1}{e-1}\cdot {\tanh }^2\frac{E}{2}}$$

and as ${\displaystyle \tan \frac{\theta }{2}}$ and ${\displaystyle \tanh \frac{E}{2}}$ have the same sign it follows that

${\displaystyle \tan \frac{\theta }{2}=\sqrt{\frac{e+1}{e-1}}\cdot \tanh \frac{E}{2}}$

(42)

This relation is convenient for passing between "true anomaly" and the parameter E, the latter being connected to time through relation (34). Note that this is a mapping between the ranges

$${\displaystyle \left[-{\cos }^{-1}\left(-\frac{1}{e}\right)<\theta <{\cos }^{-1}\left(-\frac{1}{e}\right)\right]⟷\left[-\mathrm{\infty }<E<\mathrm{\infty }\right]}$$

and that ${\displaystyle \frac{E}{2}}$ can be computed using the relation

$${\displaystyle {\tanh }^{-1}x=\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)}$$

From relation (27) follows that the orbital period P for an elliptic orbit is

${\displaystyle P=2\pi a\cdot \sqrt{\frac{a}{\alpha }}}$

(43)

As the potential energy corresponding to the force field of relation (1) is

$${\displaystyle -\frac{\alpha }{r}}$$

it follows from (13), (14), (18) and (19) that the sum of the kinetic and the potential energy

$${\displaystyle \frac{{V_r}^2+{V_t}^2}{2}-\frac{\alpha }{r}}$$

for an elliptic orbit is

${\displaystyle -\frac{\alpha }{2a}}$

(44)

and from (13), (16), (18) and (19) that the sum of the kinetic and the potential energy for a hyperbolic orbit is

${\displaystyle \frac{\alpha }{2a}}$

(45)

Relative the inertial coordinate system

$${\displaystyle \hat{x},\hat{y}}$$

in the orbital plane with ${\displaystyle \hat{x}}$ towards pericentre one gets from (18) and (19) that the velocity components are

${\displaystyle V_x=V_r\cos \theta -V_t\sin \theta =-\sqrt{\frac{\alpha }{p}}\cdot \sin \theta }$

(46)

${\displaystyle V_y=V_r\sin \theta +V_t\cos \theta =\sqrt{\frac{\alpha }{p}}\cdot (e+\cos \theta )}$

(47)

The equation of the center relates mean anomaly to true anomaly for elliptical orbits, for small numerical eccentricity.

Further information: Equation of the center § Analytical expansions

Determination of the Kepler orbit that corresponds to a given initial state

This is the "initial value problem" for the differential equation (1) which is a first order equation for the 6-dimensional "state vector" ${\displaystyle (\mathbf{r},\mathbf{v})}$ when written as

${\displaystyle \dot{\mathbf{v}}=-\alpha \cdot \frac{\hat{\mathbf{r}}}{r^2}}$

(48)

${\displaystyle \dot{\mathbf{r}}=\mathbf{v}}$

(49)

For any values for the initial "state vector" ${\displaystyle ({\mathbf{r}}_0,{\mathbf{v}}_0)}$ the Kepler orbit corresponding to the solution of this initial value problem can be found with the following algorithm:

Define the orthogonal unit vectors ${\displaystyle (\hat{\mathbf{r}},\hat{\mathbf{t}})}$ through

${\displaystyle {\mathbf{r}}_0=r\hat{\mathbf{r}}}$

(50)

${\displaystyle {\mathbf{v}}_0=V_r\hat{\mathbf{r}}+V_t\hat{\mathbf{t}}}$

(51)

with ${\displaystyle r>0}$ and ${\displaystyle V_t>0}$

From (13), (18) and (19) follows that by setting

${\displaystyle p=\frac{{(r\cdot V_t)}^2}{\alpha }}$

(52)

and by defining ${\displaystyle e\ge 0}$ and ${\displaystyle \theta }$ such that

${\displaystyle e\cos \theta =\frac{V_t}{V_0}-1}$

(53)

${\displaystyle e\sin \theta =\frac{V_r}{V_0}}$

(54)

where

${\displaystyle V_0=\sqrt{\frac{\alpha }{p}}}$

(55)

one gets a Kepler orbit that for true anomaly ${\displaystyle \theta }$ has the same r, ${\displaystyle V_r}$ and ${\displaystyle V_t}$ values as those defined by (50) and (51).

If this Kepler orbit then also has the same ${\displaystyle (\hat{\mathbf{r}},\hat{\mathbf{t}})}$ vectors for this true anomaly ${\displaystyle \theta }$ as the ones defined by (50) and (51) the state vector ${\displaystyle (\mathbf{r},\mathbf{v})}$ of the Kepler orbit takes the desired values ${\displaystyle ({\mathbf{r}}_0,{\mathbf{v}}_0)}$ for true anomaly ${\displaystyle \theta }$.

The standard inertially fixed coordinate system ${\displaystyle (\hat{\mathbf{x}},\hat{\mathbf{y}})}$ in the orbital plane (with ${\displaystyle \hat{\mathbf{x}}}$ directed from the centre of the homogeneous sphere to the pericentre) defining the orientation of the conical section (ellipse, parabola or hyperbola) can then be determined with the relation

${\displaystyle \hat{\mathbf{x}}=\cos \theta \hat{\mathbf{r}}-\sin \theta \hat{\mathbf{t}}}$

(56)

${\displaystyle \hat{\mathbf{y}}=\sin \theta \hat{\mathbf{r}}+\cos \theta \hat{\mathbf{t}}}$

(57)

Note that the relations (53) and (54) has a singularity when ${\displaystyle V_r=0}$ and

$${\displaystyle V_t=V_0=\sqrt{\frac{\alpha }{p}}=\sqrt{\frac{\alpha }{\frac{{(r\cdot V_t)}^2}{\alpha }}}}$$

i.e.

${\displaystyle V_t=\sqrt{\frac{\alpha }{r}}}$

(58)

which is the case that it is a circular orbit that is fitting the initial state ${\displaystyle ({\mathbf{r}}_0,{\mathbf{v}}_0)}$

The osculating Kepler orbit

Main article: Osculating orbit

For any state vector ${\displaystyle (\mathbf{r},\mathbf{v})}$ the Kepler orbit corresponding to this state can be computed with the algorithm defined above. First the parameters ${\displaystyle p,e,\theta }$ are determined from ${\displaystyle r,V_r,V_t}$ and then the orthogonal unit vectors in the orbital plane ${\displaystyle \hat{x},\hat{y}}$ using the relations (56) and (57).

If now the equation of motion is

${\displaystyle \ddot{\mathbf{r}}=\mathbf{F}(\mathbf{r},\dot{\mathbf{r}},t)}$

(59)

where

$${\displaystyle \mathbf{F}(\mathbf{r},\dot{\mathbf{r}},t)}$$

is a function other than

$${\displaystyle -\alpha \frac{\mathbf{r}}{r^2}}$$

the resulting parameters ${\displaystyle p}$, ${\displaystyle e}$, ${\displaystyle \theta }$, ${\displaystyle \hat{\mathbf{x}}}$, ${\displaystyle \hat{\mathbf{y}}}$ defined by ${\displaystyle \mathbf{r},\dot{\mathbf{r}}}$ will all vary with time as opposed to the case of a Kepler orbit for which only the parameter ${\displaystyle \theta }$ will vary.

The Kepler orbit computed in this way having the same "state vector" as the solution to the "equation of motion" (59) at time t is said to be "osculating" at this time.

This concept is for example useful in case

$${\displaystyle \mathbf{F}(\mathbf{r},\dot{\mathbf{r}},t)=-\alpha \frac{\hat{\mathbf{r}}}{r^2}+\mathbf{f}(\mathbf{r},\dot{\mathbf{r}},t)}$$

where

$${\displaystyle \mathbf{f}(\mathbf{r},\dot{\mathbf{r}},t)}$$

is a small "perturbing force" due to for example a faint gravitational pull from other celestial bodies. The parameters of the osculating Kepler orbit will then only slowly change and the osculating Kepler orbit is a good approximation to the real orbit for a considerable time period before and after the time of osculation.

This concept can also be useful for a rocket during powered flight as it then tells which Kepler orbit the rocket would continue in case the thrust is switched off.

For a "close to circular" orbit the concept "eccentricity vector" defined as ${\displaystyle \mathbf{e}=e\hat{\mathbf{x}}}$ is useful. From (53), (54) and (56) follows that

${\displaystyle \mathbf{e}=\frac{(V_t-V_0)\hat{\mathbf{r}}-V_r\hat{\mathbf{t}}}{V_0}}$

(60)

i.e. ${\displaystyle \mathbf{e}}$ is a smooth differentiable function of the state vector ${\displaystyle (\mathbf{r},\mathbf{v})}$ also if this state corresponds to a circular orbit.

Oort cloud

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Oort_cloud 

The distance from the Oort cloud to the interior of the Solar System, and two of the nearest stars, is measured in astronomical units. The scale is logarithmic: each indicated distance is ten times farther out than the previous distance. The red arrow indicates the location of the space probe Voyager 1 by 2012-2013, which will reach the Oort cloud in about 300 years.

 

An artist's impression of the Oort cloud and the Kuiper belt (inset); the sizes of objects are over-scaled for visibility.

The Oort cloud (/ɔːrt, ʊərt/), sometimes called the Öpik–Oort cloud, is theorized to be a vast cloud of icy planetesimals surrounding the Sun at distances ranging from 2,000 to 200,000 AU (0.03 to 3.2 light-years). The concept of such a cloud was proposed in 1950 by the Dutch astronomer Jan Oort, in whose honor the idea was named. Oort proposed that the bodies in this cloud replenish and keep constant the number of long-period comets entering the inner Solar System—where they are eventually consumed and destroyed during close approaches to the Sun.

The cloud is thought to comprise two regions: a disc-shaped inner Oort cloud aligned with the solar ecliptic (also called its Hills cloud) and a spherical outer Oort cloud enclosing the entire solar system. Both regions lie well beyond the heliosphere and are in interstellar space. The Kuiper belt, the scattered disc and the detached objects—three other reservoirs of trans-Neptunian objects—are more than a thousand times closer to the Sun than the innermost portion of the Oort cloud (as shown in a logarithmic graphic within this article).

The outer limit of the Oort cloud defines the cosmographic boundary of the Solar System. This area is defined by the Sun's Hill sphere, and hence lies at the interface between solar and galactic gravitational dominion. The outer Oort cloud is only loosely bound to the Solar System and its constituents are easily affected by the gravitational pulls of both passing stars and the Milky Way itself. These forces served to moderate and render more circular the highly eccentric orbits of material ejected from the inner solar system during its early phases of development. The circular orbits of material in the Oort disc are largely thanks to this galactic gravitational torquing. By the same token, galactic interference in the motion of Oort bodies occasionally dislodges comets from their orbits within the cloud, sending them into the inner Solar System. Based on their orbits most but not all of the short-period comets appear to have come from the Oort disc. Other short-period comets may have originated from the far larger spherical cloud.

Astronomers hypothesize that the material presently in the Oort cloud formed much closer to the Sun, in the protoplanetary disc, and was then scattered far into space through the gravitational influence of the giant planets. No direct observation of the Oort cloud is possible with present imaging technology. Nevertheless, the cloud is thought to be the source that replenishes most long-period and Halley-type comets, which are eventually consumed by their close approaches to the Sun after entering the inner Solar System. The cloud may also serve the same function for many of the centaurs and Jupiter-family comets.

Development of theory

By the turn of the 20th century it was understood that there were two main classes of comet: short-period comets (also called ecliptic comets) and long-period comets (also called nearly isotropic comets). Ecliptic comets have relatively small orbits aligned near the ecliptic plane and are not found much farther than the Kuiper cliff around 50 AU from the Sun (the orbit of Neptune averages about 30 AU and 177P/Barnard has aphelion around 48 AU). Long-period comets on the other hand travel in very large orbits thousands of AU from the Sun and are isotropically distributed. This means long-period comets appear from every direction in the sky, both above and below the ecliptic plane. The origin of these comets was not well understood and many long-period comets were initially thought to be on assumed parabolic trajectories, making them one-time visitors to the Sun from interstellar space.

In 1907 A. O. Leuschner suggested that many of the comets then thought to have parabolic orbits in fact moved along extremely large elliptical orbits that would return them to the inner Solar System after long intervals during which they were invisible to Earth-based astronomy. In 1932 the Estonian astronomer Ernst Öpik proposed a reservoir of long-period comets in the form of an orbiting cloud at the outermost edge of the Solar System. Dutch astronomer Jan Oort revived this basic idea in 1950 to resolve a paradox about the origin of comets. The following facts are not easily reconcilable with the highly elliptical orbits in which long-period comets are always found:

• Over millions and billions of years the orbits of Oort cloud comets are unstable. Celestial dynamics will eventually dictate that a comet must be pulled away by a passing star, collide with the Sun or a planet, or be ejected from the Solar System through planetary perturbations.
• Moreover, the volatile composition of comets means that as they repeatedly approach the Sun radiation gradually boils the volatiles off until the comet splits or develops an insulating crust that prevents further outgassing.

Oort reasoned that comets with orbits that closely approach the Sun cannot have been doing so since the condensation of the protoplanetary disc, more than 4.5 billion years ago. Hence long-period comets could not have formed in the current orbits in which they are always discovered and must have been held in an outer reservoir for nearly all of their existence.

Oort also studied tables of ephemera for long-period comets and discovered that there is a curious concentration of long-period comets whose farthest retreat from the Sun (their aphelia) cluster around 20,000 AU. This suggested a reservoir at that distance with a spherical, isotropic distribution. He also proposed that the relatively rare comets with orbits of about 10,000 AU probably went through one or more orbits into the inner Solar System and there had their orbits drawn inward by the gravity of the planets.

Structure and composition

The presumed distance of the Oort cloud compared to the rest of the Solar System

The Oort cloud is thought to occupy a vast space somewhere between 2,000 and 5,000 AU (0.03 and 0.08 ly) from the Sun to as far out as 50,000 AU (0.79 ly) or even 100,000 to 200,000 AU (1.58 to 3.16 ly). The region can be subdivided into a spherical outer Oort cloud with a radius of some 20,000–50,000 AU (0.32–0.79 ly) and a torus-shaped inner Oort cloud with a radius of 2,000–20,000 AU (0.03–0.32 ly).

The inner Oort cloud is sometimes known as the Hills cloud, named for Jack G. Hills, who proposed its existence in 1981. Models predict the inner cloud to be the much denser of the two, having tens or hundreds of times as many cometary nuclei as the outer cloud. The Hills cloud is thought to be necessary to explain the continued existence of the Oort cloud after billions of years.

Because it lies at the interface between the dominion of Solar and galactic gravitation, the objects composing the outer Oort cloud are only weakly bound to the Sun. This in turn allows small perturbations from nearby stars or the Milky Way itself to inject long-period (and possibly Halley-type) comets inside the orbit of Neptune. This process ought to have depleted the sparser, outer cloud and yet long-period comets with orbits well above or below the ecliptic continue to be observed. The Hills cloud is thought to be a secondary reservoir of cometary nuclei and the source of replenishment for the tenuous outer cloud as the latter's numbers are gradually depleted through losses to the inner Solar System.

The outer Oort cloud may have trillions of objects larger than 1 km (0.62 mi), and billions with diameters of 20-kilometre (12 mi). This corresponds to an absolute magnitude of more than 11.^[20] On this analysis, "neighboring" objects in the outer cloud are separated by a significant fraction of 1 AU, tens of millions of kilometres. The outer cloud's total mass is not known, but assuming that Halley's Comet is a suitable proxy for the nuclei composing the outer Oort cloud, their combined mass would be roughly 3×10²⁵ kilograms (6.6×10²⁵ lb), or five Earth masses. Formerly the outer cloud was thought to be more massive by two orders of magnitude, containing up to 380 Earth masses, but improved knowledge of the size distribution of long-period comets has led to lower estimates. No estimates of the mass of the inner Oort cloud have been published as of 2023.

If analyses of comets are representative of the whole, the vast majority of Oort-cloud objects consist of ices such as water, methane, ethane, carbon monoxide and hydrogen cyanide. However, the discovery of the object 1996 PW, an object whose appearance was consistent with a D-type asteroid in an orbit typical of a long-period comet, prompted theoretical research that suggests that the Oort cloud population consists of roughly one to two percent asteroids. Analysis of the carbon and nitrogen isotope ratios in both the long-period and Jupiter-family comets shows little difference between the two, despite their presumably vastly separate regions of origin. This suggests that both originated from the original protosolar cloud, a conclusion also supported by studies of granular size in Oort-cloud comets and by the recent impact study of Jupiter-family comet Tempel 1.

Origin

The Oort cloud is thought to have developed after the formation of planets from the primordial protoplanetary disc approximately 4.6 billion years ago. The most widely accepted hypothesis is that the Oort cloud's objects initially coalesced much closer to the Sun as part of the same process that formed the planets and minor planets. After formation, strong gravitational interactions with young gas giants, such as Jupiter, scattered the objects into extremely wide elliptical or parabolic orbits that were subsequently modified by perturbations from passing stars and giant molecular clouds into long-lived orbits detached from the gas giant region.

Recent research has been cited by NASA hypothesizing that a large number of Oort cloud objects are the product of an exchange of materials between the Sun and its sibling stars as they formed and drifted apart and it is suggested that many—possibly the majority—of Oort cloud objects did not form in close proximity to the Sun. Simulations of the evolution of the Oort cloud from the beginnings of the Solar System to the present suggest that the cloud's mass peaked around 800 million years after formation, as the pace of accretion and collision slowed and depletion began to overtake supply.

Models by Julio Ángel Fernández suggest that the scattered disc, which is the main source for periodic comets in the Solar System, might also be the primary source for Oort cloud objects. According to the models, about half of the objects scattered travel outward toward the Oort cloud, whereas a quarter are shifted inward to Jupiter's orbit, and a quarter are ejected on hyperbolic orbits. The scattered disc might still be supplying the Oort cloud with material. A third of the scattered disc's population is likely to end up in the Oort cloud after 2.5 billion years.

Computer models suggest that collisions of cometary debris during the formation period play a far greater role than was previously thought. According to these models, the number of collisions early in the Solar System's history was so great that most comets were destroyed before they reached the Oort cloud. Therefore, the current cumulative mass of the Oort cloud is far less than was once suspected. The estimated mass of the cloud is only a small part of the 50–100 Earth masses of ejected material.

Gravitational interaction with nearby stars and galactic tides modified cometary orbits to make them more circular. This explains the nearly spherical shape of the outer Oort cloud. On the other hand, the Hills cloud, which is bound more strongly to the Sun, has not acquired a spherical shape. Recent studies have shown that the formation of the Oort cloud is broadly compatible with the hypothesis that the Solar System formed as part of an embedded cluster of 200–400 stars. These early stars likely played a role in the cloud's formation, since the number of close stellar passages within the cluster was much higher than today, leading to far more frequent perturbations.

In June 2010 Harold F. Levison and others suggested on the basis of enhanced computer simulations that the Sun "captured comets from other stars while it was in its birth cluster." Their results imply that "a substantial fraction of the Oort cloud comets, perhaps exceeding 90%, are from the protoplanetary discs of other stars." In July 2020 Amir Siraj and Avi Loeb found that a captured origin for the Oort Cloud in the Sun's birth cluster could address the theoretical tension in explaining the observed ratio of outer Oort cloud to scattered disc objects, and in addition could increase the chances of a captured Planet Nine.

Comets

Comets are thought to have two separate points of origin in the Solar System. Short-period comets (those with orbits of up to 200 years) are generally accepted to have emerged from either the Kuiper belt or the scattered disc, which are two linked flat discs of icy debris beyond Neptune's orbit at 30 au and jointly extending out beyond 100 au from the Sun. Very long-period comets, such as C/1999 F1 (Catalina), whose orbits last for millions of years, are thought to originate directly from the outer Oort cloud. Other comets modeled to have come directly from the outer Oort cloud include C/2006 P1 (McNaught), C/2010 X1 (Elenin), Comet ISON, C/2013 A1 (Siding Spring), C/2017 K2, and C/2017 T2 (PANSTARRS). The orbits within the Kuiper belt are relatively stable, and so very few comets are thought to originate there. The scattered disc, however, is dynamically active, and is far more likely to be the place of origin for comets. Comets pass from the scattered disc into the realm of the outer planets, becoming what are known as centaurs. These centaurs are then sent farther inward to become the short-period comets.

There are two main varieties of short-period comet: Jupiter-family comets (those with semi-major axes of less than 5 AU) and Halley-family comets. Halley-family comets, named for their prototype, Halley's Comet, are unusual in that although they are short-period comets, it is hypothesized that their ultimate origin lies in the Oort cloud, not in the scattered disc. Based on their orbits, it is suggested they were long-period comets that were captured by the gravity of the giant planets and sent into the inner Solar System. This process may have also created the present orbits of a significant fraction of the Jupiter-family comets, although the majority of such comets are thought to have originated in the scattered disc.

Oort noted that the number of returning comets was far less than his model predicted, and this issue, known as "cometary fading", has yet to be resolved. No dynamical process is known to explain the smaller number of observed comets than Oort estimated. Hypotheses for this discrepancy include the destruction of comets due to tidal stresses, impact or heating; the loss of all volatiles, rendering some comets invisible, or the formation of a non-volatile crust on the surface. Dynamical studies of hypothetical Oort cloud comets have estimated that their occurrence in the outer-planet region would be several times higher than in the inner-planet region. This discrepancy may be due to the gravitational attraction of Jupiter, which acts as a kind of barrier, trapping incoming comets and causing them to collide with it, just as it did with Comet Shoemaker–Levy 9 in 1994. An example of a typical dynamically old comet with an origin in the Oort cloud could be C/2018 F4.

Tidal effects

Main article: Galactic tide

Most of the comets seen close to the Sun seem to have reached their current positions through gravitational perturbation of the Oort cloud by the tidal force exerted by the Milky Way. Just as the Moon's tidal force deforms Earth's oceans, causing the tides to rise and fall, the galactic tide also distorts the orbits of bodies in the outer Solar System. In the charted regions of the Solar System, these effects are negligible compared to the gravity of the Sun, but in the outer reaches of the system, the Sun's gravity is weaker and the gradient of the Milky Way's gravitational field has substantial effects. Galactic tidal forces stretch the cloud along an axis directed toward the galactic center and compress it along the other two axes; these small perturbations can shift orbits in the Oort cloud to bring objects close to the Sun. The point at which the Sun's gravity concedes its influence to the galactic tide is called the tidal truncation radius. It lies at a radius of 100,000 to 200,000 au, and marks the outer boundary of the Oort cloud.

Some scholars theorize that the galactic tide may have contributed to the formation of the Oort cloud by increasing the perihelia (smallest distances to the Sun) of planetesimals with large aphelia (largest distances to the Sun). The effects of the galactic tide are quite complex, and depend heavily on the behaviour of individual objects within a planetary system. Cumulatively, however, the effect can be quite significant: up to 90% of all comets originating from the Oort cloud may be the result of the galactic tide. Statistical models of the observed orbits of long-period comets argue that the galactic tide is the principal means by which their orbits are perturbed toward the inner Solar System.

Stellar perturbations and stellar companion hypotheses

Besides the galactic tide, the main trigger for sending comets into the inner Solar System is thought to be interaction between the Sun's Oort cloud and the gravitational fields of nearby stars or giant molecular clouds. The orbit of the Sun through the plane of the Milky Way sometimes brings it in relatively close proximity to other stellar systems. For example, it is hypothesized that 70 thousand years ago, perhaps Scholz's Star passed through the outer Oort cloud (although its low mass and high relative velocity limited its effect). During the next 10 million years the known star with the greatest possibility of perturbing the Oort cloud is Gliese 710. This process could also scatter Oort cloud objects out of the ecliptic plane, potentially also explaining its spherical distribution.

In 1984, physicist Richard A. Muller postulated that the Sun has an as-yet undetected companion, either a brown dwarf or a red dwarf, in an elliptical orbit within the Oort cloud. This object, known as Nemesis, was hypothesized to pass through a portion of the Oort cloud approximately every 26 million years, bombarding the inner Solar System with comets. However, to date no evidence of Nemesis has been found, and many lines of evidence (such as crater counts), have thrown its existence into doubt. Recent scientific analysis no longer supports the idea that extinctions on Earth happen at regular, repeating intervals. Thus, the Nemesis hypothesis is no longer needed to explain current assumptions.

A somewhat similar hypothesis was advanced by astronomer John J. Matese of the University of Louisiana at Lafayette in 2002. He contends that more comets are arriving in the inner Solar System from a particular region of the postulated Oort cloud than can be explained by the galactic tide or stellar perturbations alone, and that the most likely cause would be a Jupiter-mass object in a distant orbit. This hypothetical gas giant was nicknamed Tyche. The WISE mission, an all-sky survey using parallax measurements in order to clarify local star distances, was capable of proving or disproving the Tyche hypothesis. In 2014, NASA announced that the WISE survey had ruled out any object as they had defined it.

Future exploration

Artist's impression of the TAU spacecraft

Space probes have yet to reach the area of the Oort cloud. Voyager 1, the fastest and farthest of the interplanetary space probes currently leaving the Solar System, will reach the Oort cloud in about 300 years and would take about 30,000 years to pass through it. However, around 2025, the radioisotope thermoelectric generators on Voyager 1 will no longer supply enough power to operate any of its scientific instruments, preventing any further exploration by Voyager 1. The other four probes currently escaping the Solar System have either already stopped functioning or are predicted to stop functioning before they reach the Oort cloud.

In the 1980s, there was a concept for a probe that could reach 1,000 AU in 50 years, called TAU; among its missions would be to look for the Oort cloud.

In the 2014 Announcement of Opportunity for the Discovery program, an observatory to detect the objects in the Oort cloud (and Kuiper belt) called the "Whipple Mission" was proposed. It would monitor distant stars with a photometer, looking for transits up to 10,000 AU away. The observatory was proposed for halo orbiting around L2 with a suggested 5-year mission. It was also suggested that the Kepler observatory could have been capable of detecting objects in the Oort cloud.

Sensory processing

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Sensory_processing 

Sensory processing is the process that organizes and distinguishes sensation (sensory information) from one's own body and the environment, thus making it possible to use the body effectively within the environment. Specifically, it deals with how the brain processes multiple sensory modality inputs, such as proprioception, vision, auditory system, tactile, olfactory, vestibular system, interoception, and taste into usable functional outputs.

It has been believed for some time that inputs from different sensory organs are processed in different areas in the brain. The communication within and among these specialized areas of the brain is known as functional integration. Newer research has shown that these different regions of the brain may not be solely responsible for only one sensory modality, but could use multiple inputs to perceive what the body senses about its environment. Multisensory integration is necessary for almost every activity that we perform because the combination of multiple sensory inputs is essential for us to comprehend our surroundings.

Overview

It has been believed for some time that inputs from different sensory organs are processed in different areas in the brain, relating to systems neuroscience. Using functional neuroimaging, it can be seen that sensory-specific cortices are activated by different inputs. For example, regions in the occipital cortex are tied to vision and those on the superior temporal gyrus are recipients of auditory inputs. There exist studies suggesting deeper multisensory convergences than those at the sensory-specific cortices, which were listed earlier. This convergence of multiple sensory modalities is known as multisensory integration.

Sensory processing deals with how the brain processes sensory input from multiple sensory modalities. These include the five classic senses of vision (sight), audition (hearing), tactile stimulation (touch), olfaction (smell), and gustation (taste). Other sensory modalities exist, for example the vestibular sense (balance and the sense of movement) and proprioception (the sense of knowing one's position in space) Along with Time (The sense of knowing where one is in time or activities). It is important that the information of these different sensory modalities must be relatable. The sensory inputs themselves are in different electrical signals, and in different contexts. Through sensory processing, the brain can relate all sensory inputs into a coherent percept, upon which our interaction with the environment is ultimately based.

Basic structures involved

The different senses were always thought to be controlled by separate lobes of the brain, called projection areas. The lobes of the brain are the classifications that divide the brain both anatomically and functionally. These lobes are the Frontal lobe, responsible for conscious thought, Parietal lobe, responsible for visuospatial processing, the Occipital lobe, responsible for the sense of sight, and the temporal lobe, responsible for the senses of smell and sound. From the earliest times of neurology, it has been thought that these lobes are solely responsible for their one sensory modality input. However, newer research has shown that that may not entirely be the case.

Problems

Main article: Sensory processing disorder

See also: Sensory processing sensitivity

Sometimes there can be a problem with the encoding of the sensory information. This disorder is known as Sensory processing disorder (SPD). This disorder can be further classified into three main types.

• Sensory modulation disorder, in which patients seek sensory stimulation due to an over or under response to sensory stimuli.
• Sensory based motor disorder. Patients have incorrect processing of motor information that leads to poor motor skills.
• Sensory processing disorder or sensory discrimination disorder, which is characterized by postural control problems, lack of attentiveness, and disorganization.

There are several therapies used to treat SPD. Anna Jean Ayres claimed that a child needs a healthy "sensory diet," which is all of the activities that children engage in, that gives them the necessary sensory inputs that they need to get their brain into improving sensory processing.

History

In the 1930s, Wilder Penfield was conducting a very bizarre operation at the Montreal Neurological Institute. Penfield "pioneered the incorporation of neurophysiological principles in the practice of neurosurgery. Penfield was interested in determining a solution to solve the epileptic seizure problems that his patients were having. He used an electrode to stimulate different regions of the brain's cortex, and would ask his still conscious patient what he or she felt. This process led to the publication of his book, The Cerebral Cortex of Man. The "mapping" of the sensations his patients felt led Penfield to chart out the sensations that were triggered by stimulating different cortical regions. Mrs. H. P. Cantlie was the artist Penfield hired to illustrate his findings. The result was the conception of the first sensory Homunculus.

The Homonculus is a visual representation of the intensity of sensations derived from different parts of the body. Wilder Penfield and his colleague Herbert Jasper developed the Montreal procedure using an electrode to stimulate different parts of the brain to determine which parts were the cause of the epilepsy. This part could then be surgically removed or altered in order to regain optimal brain performance. While performing these tests, they discovered that the functional maps of the sensory and motor cortices were similar in all patients. Because of their novelty at the time, these Homonculi were hailed as the "E=mc² of Neuroscience".

Current research

There are still no definitive answers to the questions regarding the relationship between functional and structural asymmetries in the brain. There are a number of asymmetries in the human brain including how language is processed mainly in the left hemisphere of the brain. There have been some cases, however, in which individuals have comparable language skills to someone who uses his left hemisphere to process language, yet they mainly use their right or both hemispheres. These cases pose the possibility that function may not follow structure in some cognitive tasks. Current research in the fields of sensory processing and multisensory integration is aiming to hopefully unlock the mysteries behind the concept of brain lateralization.

Research on sensory processing has much to offer towards understanding the function of the brain as a whole. The primary task of multisensory integration is to figure out and sort out the vast quantities of sensory information in the body through multiple sensory modalities. These modalities not only are not independent, but they are also quite complementary. Where one sensory modality may give information on one part of a situation, another modality can pick up other necessary information. Bringing this information together facilitates the better understanding of the physical world around us.

It may seem redundant that we are being provided with multiple sensory inputs about the same object, but that is not necessarily the case. This so-called "redundant" information is in fact verification that what we are experiencing is in fact happening. Perceptions of the world are based on models that we build of the world. Sensory information informs these models, but this information can also confuse the models. Sensory illusions occur when these models do not match up. For example, where our visual system may fool us in one case, our auditory system can bring us back to a ground reality. This prevents sensory misrepresentations, because through the combination of multiple sensory modalities, the model that we create is much more robust and gives a better assessment of the situation. Thinking about it logically, it is far easier to fool one sense than it is to simultaneously fool two or more senses.

Examples

One of the earliest sensations is the olfactory sensation. Evolutionary, gustation and olfaction developed together. This multisensory integration was necessary for early humans in order to ensure that they were receiving proper nutrition from their food, and also to make sure that they were not consuming poisonous materials. There are several other sensory integrations that developed early on in the human evolutionary time line. The integration between vision and audition was necessary for spatial mapping. Integration between vision and tactile sensations developed along with our finer motor skills including better hand-eye coordination. While humans developed into bipedal organisms, balance became exponentially more essential to survival. The multisensory integration between visual inputs, vestibular (balance) inputs, and proprioception inputs played an important role in our development into upright walkers.

Audiovisual system

Perhaps one of the most studied sensory integrations is the relationship between vision and audition. These two senses perceive the same objects in the world in different ways, and by combining the two, they help us understand this information better. Vision dominates our perception of the world around us. This is because visual spatial information is one of the most reliable sensory modalities. Visual stimuli are recorded directly onto the retina, and there are few, if any, external distortions that provide incorrect information to the brain about the true location of an object. Other spatial information is not as reliable as visual spatial information. For example, consider auditory spatial input. The location of an object can sometimes be determined solely on its sound, but the sensory input can easily be modified or altered, thus giving a less reliable spatial representation of the object. Auditory information therefore is not spatially represented unlike visual stimuli. But once one has the spatial mapping from the visual information, multisensory integration helps bring the information from both the visual and auditory stimuli together to make a more robust mapping.

There have been studies done that show that a dynamic neural mechanism exists for matching the auditory and visual inputs from an event that stimulates multiple senses. One example of this that has been observed is how the brain compensates for target distance. When you are speaking with someone or watching something happen, auditory and visual signals are not being processed concurrently, but they are perceived as being simultaneous. This kind of multisensory integration can lead to slight misperceptions in the visual-auditory system in the form of the ventriloquism effect. An example of the ventriloquism effect is when a person on the television appears to have his voice coming from his mouth, rather than the television's speakers. This occurs because of a pre-existing spatial representation within the brain which is programmed to think that voices come from another human's mouth. This then makes it so the visual response to the audio input is spatially misrepresented, and therefore misaligned.

Sensorimotor system

Hand eye coordination is one example of sensory integration. In this case, we require a tight integration of what we visually perceive about an object, and what we tactilely perceive about that same object. If these two senses were not combined within the brain, then one would have less ability to manipulate an object. Eye–hand coordination is the tactile sensation in the context of the visual system. The visual system is very static, in that it does not move around much, but the hands and other parts used in tactile sensory collection can freely move around. This movement of the hands must be included in the mapping of both the tactile and visual sensations, otherwise one would not be able to comprehend where they were moving their hands, and what they were touching and looking at. An example of this happening is looking at an infant. The infant picks up objects and puts them in his mouth, or touches them to his feet or face. All of these actions are culminating to the formation of spatial maps in the brain and the realization that "Hey, that thing that's moving this object is actually a part of me." Seeing the same thing that they are feeling is a major step in the mapping that is required for infants to begin to realize that they can move their arms and interact with an object. This is the earliest and most explicit way of experiencing sensory integration.

Further research

In the future, research on sensory integration will be used to better understand how different sensory modalities are incorporated within the brain to help us perform even the simplest of tasks. For example, we do not currently have the understanding needed to comprehend how neural circuits transform sensory cues into changes in motor activities. More research done on the sensorimotor system can help understand how these movements are controlled. This understanding can potentially be used to learn more about how to make better prosthetics, and eventually help patients who have lost the use of a limb. Also, by learning more about how different sensory inputs can combine can have profound effects on new engineering approaches using robotics. The robot's sensory devices may take in inputs of different modalities, but if we understand multisensory integration better, we might be able to program these robots to convey these data into a useful output to better serve our purposes.