Main roads

In Geometry, the main roads of a ellipse is a parameter used to describe the dimension of this Conique. The equatorial radius is half of the main roads.

Definition

The main roads of a ellipse are its larger Diamètre, a segment which crosses at the same time the center and both hearths of the ellipse and joined it in its two opposite points. The equatorial radius corresponds to half of the main roads and joint the center and an edge of the ellipse through one of the hearths.

In a similar way, the segment Perpendicular to the main roads, passing by the center and joining the ellipse is its Petit axis. The axes are the elliptic equivalents of the diameters of a Cercle, while the semi-axes are the analogues of the rays.

The length of the equatorial radius $a$ and that of the half-small axis $b$ are bound by the Excentricité $e$ and the parameter $\backslash \; ell$:

$b\; =\; has\; \backslash \; sqrt\; \{1-e^2\}$

$\backslash \; ell=a\; (1-e^2)$

$a\; \backslash \; ell=b^2$

Astronomy

Orbital period

In Astronomy, the equatorial radius is a orbital element important, making it possible to define a Orbite partially. In a general way, within the framework of a Problem to two bodies, the orbital Period of a body of mass $m$ orbiting around an other body of mass $M$ is:

$T^2=\; \backslash \; frac\; \{4\; \backslash \; pi^2\}\; \{G\; (M+m)\}a^3\; \backslash ,$

where:

• $a$ is the length of the equatorial radius
• $G$ is the gravitational Constante.

If one of the bodies is sufficiently small so that its mass is neglected compared to the other:

$T\; =\; 2\; \backslash \; pi\; \backslash \; sqrt\; \{a^3/\backslash \; driven\}$

where $\backslash \; mu$ is the gravitational Paramètre standard.

Of this case, for all the orbits in the same way equatorial radius, the period is same whatever the eccentricity.

The following proportionality is thus obtained:

$T^2\; \backslash propto\; a^3$

what corresponds to the third law of Kepler.

Outdistance average

The equatorial radius does not correspond inevitably to the average distance between the two bodies in orbit, because this distance depends on the process used:

• By calculating the average of the distance on the Eccentric anomaly, one finds indeed the equatorial radius.

• the average on the True anomaly gives the Half-small axis.
• Moyenner on the Mean anomaly leads to the value $a\; (1\; +\; \backslash \; frac\; \{e^2\}\; \{2\})$.

In addition, the average radius of the ellipse, measured compared to its center, is $\backslash \; sqrt\; \{ab\}\; =\; has\; \backslash \; sqrt\; \{1-e^2\}$.

See too

• orbital Elements:
• Mean anomaly
• Argument of the pericenter
• orbital Eccentricity
• Slope
• Longitude of the ascending node

Simple: Semi-major axis

Random links:

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