Elongation (astronomy)
In astronomy, the elongation is the angle between the Sun and a planet seen from the Earth. For an inferior planet like Mercury and Venus, the elongation acquires a maximum value given by the expression
- without Emax=r{displaystyle sin E_{max}=r}
Where r is the distance of the inferior planet from the Sun in AU. For a higher planet the elongation does not have a limited value. It is worth 0 in the conjunction, 90 in the squares and 180 in the opposition.
At the maximum elongation of a lower planet, the phase angle, which is by definition the angle formed by the Planet-Sun segment with the Planet-Earth segment, is 90º and it is said that the planet presents " dichotomy" since from the Earth one can observe half an illuminated circle and a dark half circle, just as we see the Moon in the quarters.
When an inferior planet is visible after sunset, it is near its maximum eastern elongation. When an inferior planet is visible before dawn, it is near its maximum western elongation. The value of the maximum elongation (west or east), for Mercury, is between 18° and 28° and for Venus between 45° and 47°. This value varies because the orbits of the planets are elliptical, rather than perfect circles. Another minor contributor to this inconsistency is orbital inclination: each planet's orbit is in a slightly different plane.
Elongation is also understood as the angle between a planet and its satellite seen from Earth. Galileo studied the changing configurations of Jupiter's satellites by measuring their elongations.
Period between maximum elongations
The maximum elongations of an inferior planet occur periodically, with a maximum eastern elongation followed by a maximum western elongation, and vice versa. The period depends on the relative angular velocity of the Earth and the planet as seen from the Sun. The time it takes to complete it is the synodic period of the planet.
If Ts is the synodic period (for example, the interval between two maximum eastern elongations, or the interval between two maximum superior conjunctions,...), ω the relative angular velocity, ωe the angular velocity of the Earth, and ωp that of the planet, then we have that for a lower planet (Mercury, Venus)
- Ts=2π π ω ω =2π π ω ω p− − ω ω e=2π π 2π π Tp− − 2π π Te=TeTeTp− − 1{displaystyle T_{s}={2pi over omega }={2pi over omega _{p}-omega _{e}}={2pi over {2pi over T_{p}}}-{2pi over T_{e}}}}={T_{e} over {T_{e} over T_{p}}
- Ts=TeTpTe− − Tp{displaystyle T_{s}={dfrac {T_{e}T_{p}{T_{p}}}}}}}}
While for a higher planet (Mars, Jupiter,...) in which there is no maximum elongation, since the elongation can take all values from 0º to 180º, the synodic period, time between, for example, two superior conjunctions (elongation = 0º) or two consecutive oppositions (elongation = 180º), it is calculated:
- Ts=2π π ω ω =2π π ω ω e− − ω ω p=2π π 2π π Te− − 2π π Tp=Te1− − TeTp{displaystyle T_{s}={2pi over omega }={2pi over omega _{e}-omega _{p}}}={2pi over {2pi over T_{e}}}{2pi over T_{p}}}}{dfrac {T_{ep}{1}{1⁄2}{1⁄4}}{ep}}}}}}}}}}{1⁄4}}{1⁄4}}}{
- Ts=TeTpTp− − Te{displaystyle T_{s}={dfrac {T_{e}{T_{p}{T_{e}}}}}}}}
Where Te and Tp are the "years&# 3. 4; of the Earth and the planet respectively, that is to say the periods of revolution around the Sun, called the sidereal period.
For example, Venus' sidereal period (p) is 225 days, and Earth's (e) is 365 days. Thus, the synodic period (s) of Venus, which provides the lapse between two maximum eastern (or western) elongations, is 584 days.
These values are approximate, because the planets do not have perfectly circular coplanar orbits. As Kepler's second law dictates, when a planet is close to the Sun it moves faster than when it is farther away, so determining the exact date and time of greatest elongation requires a much more complex analysis of orbital mechanics.
Elongation of superior planets
Superior planets, dwarf planets, and asteroids have different cycles. After superior conjunction, the elongation of the object continues to increase until it approaches a maximum value greater than 90° (which is impossible with inferior planets) and typically very close to 180°, known as opposition and corresponds to the heliocentric conjunction with the Earth. In other words, as seen by an observer on the uppermost planet at opposition, Earth appears inferiorly conjunct the Sun. Technically, the exact moment of opposition is a little different from the moment of greatest elongation. Opposition is defined as the moment when the apparent ecliptic longitudes of the upper planet and the Sun differ by 180°, ignoring the fact that the planet is out of the plane of Earth's orbit. For example, Pluto, whose orbit is highly inclined to Earth's orbital plane, may have a maximum elongation significantly less than 180° at opposition.
All of the upper planets are most easily visible at their oppositions because they are near their closest approach to Earth and are also above the horizon all night. The variation in magnitude caused by changes in elongation is greater the closer the planet's orbit is to Earth. In particular the magnitude of Mars changes up to 75 times with elongation. Jupiter's maximum and minimum brightness differ by only a factor of 3.3 times, while that of Uranus—which is the most distant body in the Solar System visible to the naked eye—differs by a factor of 1.7 times.
Since asteroids travel in an orbit not much larger than Earth's, their magnitude can vary greatly as a function of elongation. Although more than a dozen objects in the asteroid belt can be seen with 10x50 binoculars at average opposition, only Ceres and Vesta are always above the binocular limit of +9.5 at small elongations.
Elongation of the satellites of other planets
Sometimes, elongation can refer to the angular distance of a satellite from its central planet, such as Io's angular distance from Jupiter. Here one can also speak of greater eastern elongations and greater western elongations. In the case of Uranus' satellites, we can speak of maximum North elongation and maximum South elongation, due to the high inclination of the planet's axis of rotation.