Semi-major and semi-minor axes

Hyperbola

The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches; if this is a in the x-direction the equation is:

${\displaystyle \frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1.}$

In terms of the semi-latus rectum and the eccentricity, we have

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

The transverse axis of a hyperbola coincides with the major axis.

In a hyperbola, a conjugate axis or minor axis of length ${\displaystyle 2b}$, corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints ${\displaystyle (0,\pm b)}$ of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length b. Denoting the semi-major axis length (distance from the center to a vertex) as a, the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows:

${\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1.}$

The semi-minor axis is also the distance from one of focuses of the hyperbola to an asymptote. Often called the impact parameter, this is important in physics and astronomy, and measure the distance a particle will miss the focus by if its journey is unperturbed by the body at the focus.

The semi-minor axis and the semi-major axis are related through the eccentricity, as follows:

${\displaystyle b=a\sqrt{e^2-1}.}$

Note that in a hyperbola b can be larger than a.

Astronomy

Orbital period

In astrodynamics the orbital period T of a small body orbiting a central body in a circular or elliptical orbit is:

${\displaystyle T=2\pi \sqrt{\frac{a^3}{\mu }},}$

where:

a is the length of the orbit's semi-major axis, ${\displaystyle \mu }$ is the standard gravitational parameter of the central body.

Note that for all ellipses with a given semi-major axis, the orbital period is the same, disregarding their eccentricity.

The specific angular momentum h of a small body orbiting a central body in a circular or elliptical orbit is

${\displaystyle h=\sqrt{a\mu (1-e^2)},}$

where:

a and ${\displaystyle \mu }$ are as defined above, e is the eccentricity of the orbit.

In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For Solar System objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived):

${\displaystyle T^2\propto a^3,}$

where T is the period, and a is the semi-major axis. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton:

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

where G is the gravitational constant, M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered.

The orbiting body's path around the barycenter and its path relative to its primary are both ellipses. The semi-major axis is sometimes used in astronomy as the primary-to-secondary distance when the mass ratio of the primary to the secondary is significantly large (${\displaystyle M\gg m}$); thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth–Moon system. The mass ratio in this case is 81.30059. The Earth–Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. (Given the lunar orbit's eccentricity e = 0.0549, its semi-minor axis is 383,800 km. Thus the Moon's orbit is almost circular.) The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,730 km, the Earth's counter-orbit taking up the difference, 4,670 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives a geocentric lunar average orbital speed of 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.

Average distance

It is often said that the semi-major axis is the "average" distance between the primary focus of the ellipse and the orbiting body. This is not quite accurate, because it depends on what the average is taken over. The time- and angle-averaged distance of the orbiting body can vary by 50-100% from the orbital semi-major axis, depending on the eccentricity.

• averaging the distance over the eccentric anomaly indeed results in the semi-major axis.
• averaging over the true anomaly (the true orbital angle, measured at the focus) results in the semi-minor axis ${\displaystyle b=a\sqrt{1-e^2}}$.
• averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle) gives the time-average ${\displaystyle a\left(1+\frac{e^2}{2}\right)}$.

The time-averaged value of the reciprocal of the radius, ${\displaystyle r^{-1}}$, is ${\displaystyle a^{-1}}$.

Energy; calculation of semi-major axis from state vectors

In astrodynamics, the semi-major axis a can be calculated from orbital state vectors:

${\displaystyle a=-\frac{\mu }{2\epsilon }}$

for an elliptical orbit and, depending on the convention, the same or

${\displaystyle a=\frac{\mu }{2\epsilon }}$

for a hyperbolic trajectory, and

${\displaystyle \epsilon =\frac{v^2}{2}-\frac{\mu }{|\mathbf{r}|}}$

(specific orbital energy) and

${\displaystyle \mu =GM,}$

(standard gravitational parameter), where:

v is orbital velocity from velocity vector of an orbiting object,

r is a cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),

G is the gravitational constant,

M is the mass of the gravitating body, and

${\displaystyle \epsilon }$ is the specific energy of the orbiting body.

Note that for a given amount of total mass, the specific energy and the semi-major axis are always the same, regardless of eccentricity or the ratio of the masses. Conversely, for a given total mass and semi-major axis, the total specific orbital energy is always the same. This statement will always be true under any given conditions.

Semi-major and semi-minor axes of the planets' orbits

Planet orbits are always cited as prime examples of ellipses (Kepler's first law). However, the minimal difference between the semi-major and semi-minor axes shows that they are virtually circular in appearance. That difference (or ratio) is based on the eccentricity and is computed as ${\displaystyle \frac{a}{b}=\frac{1}{\sqrt{1-e^2}}}$, which for typical planet eccentricities yields very small results.

The reason for the assumption of prominent elliptical orbits lies probably in the much larger difference between aphelion and perihelion. That difference (or ratio) is also based on the eccentricity and is computed as ${\displaystyle \frac{r_{\text{a}}}{r_{\text{p}}}=\frac{1+e}{1-e}}$. Due to the large difference between aphelion and perihelion, Kepler's second law is easily visualized.

1 AU (astronomical unit) equals 149.6 million km.