Chord (astronomy)
Encyclopedia
In the field of astronomy
the term chord typically refers to a line crossing an object which is formed during an occultation
event. By taking accurate measurements of the start and end times of the event, in conjunction with the known location of the observer and the object's orbit
, the length of the chord can be determined giving an indication of the size of the occulting object. By combining observations made from several different locations, multiple chords crossing the occulting object can be determined giving a more accurate shape and size model. This technique of using multiple observers during the same event has been used to derive more sophisticated shape models for asteroid
s, whose shape can be highly irregular. A notable example of this occurred in 2002 when the asteroid 345 Tercidina
underwent a stellar occultation of a very bright star as seen from Europe
. During this event a team of at least 105 observers recorded 75 chords across the asteroid's surface allowing for a very accurate size and shape determination.
In addition to using a known orbit to determine an objects size, the reverse process can also be used. In this usage the occulting object's size is taken to be known and the occultation time can be used to determine the length of the chord the background object traced across the foreground object. Knowing this chord and the foreground object's size, a more precise orbit for the object can be determined.
This usage of the term "chord" is similar to the geometric concept (see: Chord (geometry)
). The difference being that in the geometric sense a chord refers to a line segment whose ends lie on a circle, whereas in the astronomical sense the occulting shape is not necessarily circular making the astonomical usage of the term more akin to a secant line
.
and the star's lightcurve is recorded. The recording of the lightcurve continues during and for a short time after the predicted event. This extra recording time is due in part to uncertainties in the occulting objects orbit but also due to the possibility of detecting other objects orbiting the primary object (for example in the case of a binary asteroid
, also the ring
system around the planet Uranus
was detected this way).
The exact method of lightcurve determination is dependent on the specific equipment available to the observer and the goals of the observation, however in all occultation events accurate timing is an essential component of the observation process. The exact time that the foreground object eclipses the other can be used to work out a very precise position along the occulting object's orbit. Also, since the duration of the drop in the measured lightcurve gives the object's size and since occultation events typically only last somewhere on the order of a few seconds, very fast integration times
are required to allow for high temporal resolution
along the lightcurve. A second method of achieving very high temporal accuracy is to actually use a long exposure and allow the target star to drift across the CCD during the exposure. This method, known as the trailed image method, produces a streak along the photograph whose thickness corresponds to the brightness of the target star with the distance along the streak direction indicates time; this allows for very high temporal accuracy even when the target star may be too dim for the method described above using high frequency short exposures. With high enough temporal resolution even the angular size of the background star can be determined.
Once the lightcurve has been recorded the chord across the occulting object can be determined via calculation. By using the start and end times of the occultation event the position in space of both the observer and the occulting object can be worked out (a process complicated by the fact that both the object and the observer are moving). Knowing these two locations, combined with the direction to the background object, the two endpoints of the chord can be determined using simple geometry
.
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
the term chord typically refers to a line crossing an object which is formed during an occultation
Occultation
An occultation is an event that occurs when one object is hidden by another object that passes between it and the observer. The word is used in astronomy . It can also refer to any situation wherein an object in the foreground blocks from view an object in the background...
event. By taking accurate measurements of the start and end times of the event, in conjunction with the known location of the observer and the object's orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
, the length of the chord can be determined giving an indication of the size of the occulting object. By combining observations made from several different locations, multiple chords crossing the occulting object can be determined giving a more accurate shape and size model. This technique of using multiple observers during the same event has been used to derive more sophisticated shape models for asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...
s, whose shape can be highly irregular. A notable example of this occurred in 2002 when the asteroid 345 Tercidina
345 Tercidina
345 Tercidina is a large main-belt asteroid. It is classified as a C-type asteroid and is probably composed of carbonaceous material.It was discovered by Auguste Charlois on November 23, 1892, in Nice.-Size:Via asteroid occultations:...
underwent a stellar occultation of a very bright star as seen from Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...
. During this event a team of at least 105 observers recorded 75 chords across the asteroid's surface allowing for a very accurate size and shape determination.
In addition to using a known orbit to determine an objects size, the reverse process can also be used. In this usage the occulting object's size is taken to be known and the occultation time can be used to determine the length of the chord the background object traced across the foreground object. Knowing this chord and the foreground object's size, a more precise orbit for the object can be determined.
This usage of the term "chord" is similar to the geometric concept (see: Chord (geometry)
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
). The difference being that in the geometric sense a chord refers to a line segment whose ends lie on a circle, whereas in the astronomical sense the occulting shape is not necessarily circular making the astonomical usage of the term more akin to a secant line
Secant line
A secant line of a curve is a line that intersects two points on the curve. The word secant comes from the Latin secare, to cut.It can be used to approximate the tangent to a curve, at some point P...
.
Observation process
Because an occultation event for an individual object is quite rare, the process of observing occultation events begins with the creation of a list of candidate targets. The list is generated from a computer by analyzing the orbital motions of a large collection of objects with known orbital parameters. Once a candidate event has been chosen whose ground track passes over the site of an observer, the preparations for the observation begin. A few minutes before the event is expected to happen the observing telescope is targeted to the target starStar
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
and the star's lightcurve is recorded. The recording of the lightcurve continues during and for a short time after the predicted event. This extra recording time is due in part to uncertainties in the occulting objects orbit but also due to the possibility of detecting other objects orbiting the primary object (for example in the case of a binary asteroid
Binary asteroid
A binary asteroid is a system of two asteroids orbiting their common center of mass, in analogy with binary stars. 243 Ida was the first binary asteroid to be identified when the Galileo spacecraft did a flyby in 1993...
, also the ring
Planetary ring
A planetary ring is a ring of cosmic dust and other small particles orbiting around a planet in a flat disc-shaped region.The most notable planetary rings known in Earth's solar system are those around Saturn, but the other three gas giants of the solar system possess ring systems of their...
system around the planet Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...
was detected this way).
The exact method of lightcurve determination is dependent on the specific equipment available to the observer and the goals of the observation, however in all occultation events accurate timing is an essential component of the observation process. The exact time that the foreground object eclipses the other can be used to work out a very precise position along the occulting object's orbit. Also, since the duration of the drop in the measured lightcurve gives the object's size and since occultation events typically only last somewhere on the order of a few seconds, very fast integration times
Shutter speed
In photography, shutter speed is a common term used to discuss exposure time, the effective length of time a camera's shutter is open....
are required to allow for high temporal resolution
Temporal resolution
Temporal resolution refers to the precision of a measurement with respect to time. Often there is a tradeoff between temporal resolution of a measurement and its spatial resolution. This trade-off can be attributed to the finite speed of light and the fact that it takes a certain period of time...
along the lightcurve. A second method of achieving very high temporal accuracy is to actually use a long exposure and allow the target star to drift across the CCD during the exposure. This method, known as the trailed image method, produces a streak along the photograph whose thickness corresponds to the brightness of the target star with the distance along the streak direction indicates time; this allows for very high temporal accuracy even when the target star may be too dim for the method described above using high frequency short exposures. With high enough temporal resolution even the angular size of the background star can be determined.
Once the lightcurve has been recorded the chord across the occulting object can be determined via calculation. By using the start and end times of the occultation event the position in space of both the observer and the occulting object can be worked out (a process complicated by the fact that both the object and the observer are moving). Knowing these two locations, combined with the direction to the background object, the two endpoints of the chord can be determined using simple geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
.