Gamma (eclipse)

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Gamma of a central total eclipse. This illustrates solar eclipse gamma: The red line shows the least distance from the center of the Earth, in this case approximately 75% of radius of the Earth. Because the umbra passes north of the Earth's center, gamma in this example is +0.75.

Gamma (denoted as γ) of an eclipse describes how centrally the shadow of the Moon or Earth strikes the other body. This distance, measured at the moment when the axis of the shadow cone passes closest to the center of the Earth or Moon, is stated as a fraction of the equatorial radius of the Earth or Moon.

Sign[edit]

The sign of gamma defines, for a solar eclipse, if the axis of the shadow passes north or south of the center of the Earth; a positive value means north. The Earth is defined as that half which is exposed to the Sun (this changes with the seasons and is not related directly to the Earth's poles or equator; thus, the Earth's center is wherever the Sun is directly overhead).

For a lunar eclipse, it defines whether the axis of the Earth's shadow passes north or south of the Moon; a positive value means south.

Gamma changes monotonically throughout any single saros series. The change in gamma is larger when Earth is near its aphelion (June to July) than when it is near perihelion (December to January). For odd numbered series (ascending node for solar eclipses and descending node for lunar eclipses), gamma decreases for solar eclipses and gamma increases for lunar eclipses, while for even numbered series (descending node for solar eclipses and ascending node for lunar eclipses), gamma increases for solar eclipses and gamma decreases for lunar eclipses. This simple rule describes the current behavior of gamma, but this has not always been the case. The eccentricity of Earth's orbit is presently 0.0167, and is slowly decreasing. It was 0.0181 in the year -2000 and will be 0.0163 in +3000. In the past, when the eccentricity was larger, there were saros series in which the trend in gamma reversed for one or more saros cycles before resuming its original direction. These instances occur near perihelion when the Sun's apparent motion is highest and may, in fact, overtake the eastward shift of the node. The resulting effect is a relative shift west of the node after one saros cycle instead of the usual eastward shift. Consequently, gamma reverses direction.

Magnitude for the parameter gamma[edit]

The absolute value of gamma (denoted as |γ|) distinguishes between different types of solar eclipses that are seen on Earth.[1]

If the Earth were a perfect sphere, the maximum value of |γ| that a central eclipse[α] could have would be 1.0. However, due to the oblateness of the Earth (which causes the length of the Earth's polar radius to be about 20 km shorter than the equatorial radius), the maximum value of |γ| that a central eclipse could have is 0.9972.[2]

|γ| Type of eclipse Notes
0 Central[α] The axis of the shadow cone is exactly between the northern and southern halves of the sunlit side of the Earth
between 0 and 0.9677826 Central[α]
between 0.9677826 and 0.9972 Central[α] (one limit) One edge of the shadow cone misses the Earth
between 0.9972 and 1.0266174 non-central The axis of the shadow cone misses Earth, but, portions of the umbra or antumbra do intersect the Earth (generally in the polar regions) given the shadow cone's nonzero width.
between 1.0266174 and approximately 1.55[β] Partial
Greater than approximately 1.55[β] No eclipse
  1. ^ a b c d Defined as locations on Earth exist in which the Moon can be seen centered in front of the Sun. The eclipse can either be total or annual depending on the exact distance separating the earth from the moon (total eclipse occurs for longer distances, annular eclipse occurs for shorter distances)
  2. ^ a b The exact value depends on the exact distance separating the earth from the moon. When the separation is at its greatest, the exact maximum value for |γ| would be about 1.53, and at the smallest separation distance, the exact maximum value for |γ| would be 1.57. See
    J. Meeus: Mathematical Astronomy, Morsels, Willmann-Bell, 2000, ISBN 0-943396-51-4, Fig. 10.c. and
    J. Meeus: Mathematical Astronomy, Morsels III, Willmann-Bell, 2004, ISBN 0-943396-81-6, Page 46

The Solar eclipse of April 29, 2014, with a gamma of -0.99996, is an example of the special case of a non-central annular eclipse. The axis of the shadow cone barely missed Earth's south pole. Thus, no central line could be specified for the zone of annular visibility.[3]

The next non-central eclipse in 21st century is total solar eclipse of April 9, 2043.

Limiting cases for lunar eclipses on the Moon with respect to Earth's umbral and penumbral shadows[edit]

There are three types of lunar eclipses:

  • Penumbral Lunar Eclipse = The Moon passes through Earth's penumbra, but the Earth's umbra misses the Moon.
  • Partial Lunar Eclipse = The Moon passes through Earth's umbra, but not completely.
  • Total Lunar Eclipse = The Moon passes through Earth's umbra, completely.

The gamma is the limit of:

  • If |γ| is 0, the Moon's center passes exactly through the axis of the Earth's umbra.
  • If |γ| is lower than 0.2725, this lunar eclipse is central.
  • If |γ| is between 0.2725 and 0.47, this lunar eclipse is total.
  • If |γ| is between 0.43 and 0.987, this lunar eclipse is partial with a penumbral magnitude greater than 1.
  • If |γ| is between 0.987 and 1.0266174, this lunar eclipse is total penumbral or partial with a penumbral magnitude less than 1.
  • If |γ| is between 1.0266174 and approximately 1.55, this lunar eclipse is penumbral with penumbral magnitude less than 1, the Moon traverses only the Earth's penumbra.
  • If |γ| exceeds approximately 1.55, the Earth's penumbra misses the Moon completely, and no eclipse occurs.

References[edit]

  1. ^ J. Meeus: Astronomical Algorithms. 2nd ed., Willmann-Bell, Richmond 2000, ISBN 0-943396-61-1, Chapter 54
  2. ^ J. Meeus: Mathematical Astronomy Morsels III. Willmann-Bell, Richmond 2004, ISBN 0-943396-81-6, Chapter 6
  3. ^ Fred Espenak: Path of the Annular Solar Eclipse of 2014 Apr 29