Welfare cost of business cycles

2013-11-25T04:30:19Z

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[[File:Beta angle sun.svg|400px|thumb|Beta Angle (<math>\boldsymbol{\beta}</math>)]]
The '''beta angle''' (<math>\boldsymbol{\beta}</math>) is a measurement that is used most notably in [[spaceflight]]. The beta angle determines the percentage of time an object such as a spacecraft in [[low Earth orbit]] (LEO) spends in direct sunlight, absorbing solar energy.<ref name="mcc">{{cite web|url=http://spaceflight.nasa.gov/feedback/expert/answer/mcc/sts-113/11_23_20_01_179.html|title=MCC Answers|author=Derek Hassman, NASA Flight Director|date=December 1, 2002|publisher=NASA|accessdate=June 14, 2009}}</ref> Beta angle is defined as the angle between the [[orbit]] plane and the [[Euclidean vector|vector]] from the [[sun]] (which direction the sun is shining from).<ref name="knk">{{cite web|url=http://www.tak2000.com/data/planets/earth.htm|title= Earth's Thermal Environment|last=K&K Associates|year=2008|work=Thermal Environments JPL D-8160|publisher=K&K Associates|accessdate=July 14, 2009}}</ref> The beta angle is the smaller angle (there are two angles) between the sun vector (where the sun is shining from in the sky) and the [[Plane (geometry)|plane]] of the object's orbit. Note that the beta angle does not define a unique orbit plane; all satellites in orbit with a given beta angle at a given altitude have the same exposure to the sun, even though they may be orbiting in completely different planes around the Earth.<ref name="orbit"/> The beta angle varies between +90° and −90°, and the direction the satellite revolves around the body it orbits determines whether the beta angle sign is positive or negative. An imaginary observer standing on the sun defines a beta angle as positive if the satellite in question orbits in a counter clockwise direction and negative if it revolves clockwise.<ref name="orbit">{{cite web|url=http://wwwiutp.univ-poitiers.fr/COURS-DOCS/docu/ideas/SDRCHelp/LANG/French/tmg_ug/orbit_def.htm|title= Orbit Definition|publisher=2001 Structural Dynamics Research Corporation| accessdate=August 26, 2009}}</ref> The maximum amount of time that a satellite in a normal low Earth orbit mission can spend in the Earth's shadow occurs at a beta angle of zero. In such an orbit, the satellite is in sunlight no less than 59% of the time.<ref name="mcc"/><ref name="knk"/>

==Light and shadow==
The degree of orbital shadowing an object in LEO experiences is determined by that object's beta angle. An object launched into an initial orbit with an inclination equivalent to the complement to the Earth's inclination to the ecliptic results in an initial beta angle of 0 degrees (<math>\beta</math> = 0°) for the orbiting object. This allows the object to spend the maximum possible amount of its [[orbital period]] in the Earth's shadow, and results in extremely reduced absorption of solar energy. At a LEO of 280 kilometers the object in orbit is in sunlight through 59% of its orbit (approximately 53 minutes in sunlight, and 37 minutes in shadow.<ref name="knk"/>) On the other extreme, an object launched into an orbit that follows the [[Terminator (solar)|terminator]] results in a beta angle of 90 degrees (<math>\beta</math> = 90°), and the object is in sunlight 100% of the time.<ref name="knk"/> An example would be a polar orbit initiated at local dawn or dusk on an [[equinox]]. These orbits can be taken advantage of to keep a satellite as cool as possible for instruments that require low temperatures, such as infrared cameras, by keeping the beta angle as close to zero as possible, or conversely to keep a satellite in sunlight as much as possible for conversion of sunlight by its solar panels, for solar stability of sensors, or to study the sun by maintaining a beta angle as close to +90 or -90 as possible.

==Determination and Application of beta angles==

The above discussion defines the beta angle of satellites orbiting the Earth, but a beta angle can be calculated for any orbiting three body system: the same definition can be applied to give the beta angle of other objects. For example, the beta angle of a satellite in orbit around Mars, with respect to the Earth, defines how much of the time the satellite has a line of sight to the Earth - that is, it determines how long the Earth is shining on the satellite and how long the Earth is blocked from view. That same satellite also will have a beta angle with respect to the Sun, and in fact it has a beta angle for any celestial object one might wish to calculate one for: any satellite orbiting a body (i.e. the Earth) will be in that body's shadow with respect to a given celestial object (like a star) some of the time, and in its line-of-sight the rest of the time. Beta angles describing non-[[geocentric]] orbits are important when space agencies launch satellites into orbits around other bodies in the solar system.

==Importance in Spaceflight==

When the [[space shuttle]] was in service on missions to the [[International Space Station]], the beta angle of the space station's orbit was a crucial consideration; periods referred to as "beta cutout",<ref name="mcc"/> during which the shuttle could not safely be launched to the ISS, were a direct result of the beta angle of the space station at those times. When the orbiter was in-flight (not docked to ISS) and it flew to a beta angle greater than 60 degrees, the orbiter went into "rotisserie" mode, and slowly rotated around its X-axis (nose to tail axis). For flights to ISS, the shuttle could launch during an ISS beta cutout if the ISS would be at a beta less than 60 degrees at dock, and throughout the docked phase.{{Citation needed|date=June 2009}} Therefore, the mission duration affected launch timing when the beta cutout dates were approaching.

==See also==
*[[International Space Station]]
*[[Low Earth Orbit]]
*[[Launch window]]

==References==
{{Reflist}}

==External links==
*[http://spaceflight.nasa.gov/station/flash/start.swf NASA: ISS Beta Angle]

{{DEFAULTSORT:Beta Angle}}
[[Category:Spaceflight]]
[[Category:Earth orbits]]
[[Category:Astrodynamics]]

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Welfare cost of business cycles