Misplaced Pages

#519480

43-503: A celestial body , as the sun or moon or an object that gives light ; or, a person of eminence or brilliant achievement. From Old French luminarie or late Latin luminarium , from Latin lumen , lumin- "light". Luminary may also refer to: Celestial body Examples of astronomical objects include planetary systems , star clusters , nebulae , and galaxies , while asteroids , moons , planets , and stars are astronomical bodies. A comet may be identified as both

86-479: A 3 . {\displaystyle n={\sqrt {\frac {\mu }{a^{3}}}}.} where μ is the standard gravitational parameter . Hence if at any instant t 0 the orbital parameters are ( e 0 , a 0 , i 0 , Ω 0 , ω 0 , M 0 ) , then the elements at time t = t 0 + δt is given by ( e 0 , a 0 , i 0 , Ω 0 , ω 0 , M 0 + n δt ) . Unperturbed, two-body , Newtonian orbits are always conic sections , so

129-427: A supermassive black hole , which may result in an active galactic nucleus . Galaxies can also have satellites in the form of dwarf galaxies and globular clusters . The constituents of a galaxy are formed out of gaseous matter that assembles through gravitational self-attraction in a hierarchical manner. At this level, the resulting fundamental components are the stars, which are typically assembled in clusters from

172-455: A variable star . An example of this is the instability strip , a region of the H-R diagram that includes Delta Scuti , RR Lyrae and Cepheid variables . The evolving star may eject some portion of its atmosphere to form a nebula, either steadily to form a planetary nebula or in a supernova explosion that leaves a remnant . Depending on the initial mass of the star and the presence or absence of

215-406: A body and an object: It is a body when referring to the frozen nucleus of ice and dust, and an object when describing the entire comet with its diffuse coma and tail . Astronomical objects such as stars , planets , nebulae , asteroids and comets have been observed for thousands of years, although early cultures thought of these bodies as gods or deities. These early cultures found

258-542: A companion, a star may spend the last part of its life as a compact object ; either a white dwarf , neutron star , or black hole . The IAU definitions of planet and dwarf planet require that a Sun-orbiting astronomical body has undergone the rounding process to reach a roughly spherical shape, an achievement known as hydrostatic equilibrium . The same spheroidal shape can be seen on smaller rocky planets like Mars to gas giants like Jupiter . Any natural Sun-orbiting body that has not reached hydrostatic equilibrium

301-440: A non-inertial frame centered on one of the bodies, only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body (usually the most massive) is called the primary , the other body is called the secondary . The primary does not necessarily possess more mass than

344-400: A real geometric angle, but rather varies linearly with time, one whole orbital period being represented by an "angle" of 2 π radians . It can be converted into the true anomaly ν , which does represent the real geometric angle in the plane of the ellipse, between periapsis (closest approach to the central body) and the position of the orbiting body at any given time. Thus, the true anomaly

387-524: A two-line element: The Delaunay orbital elements were introduced by Charles-Eugène Delaunay during his study of the motion of the Moon . Commonly called Delaunay variables , they are a set of canonical variables , which are action-angle coordinates . The angles are simple sums of some of the Keplerian angles: along with their respective conjugate momenta , L , G , and H . The momenta L , G , and H are

430-426: A web that spans the observable universe. Galaxies have a variety of morphologies , with irregular , elliptical and disk-like shapes, depending on their formation and evolutionary histories, including interaction with other galaxies, which may lead to a merger . Disc galaxies encompass lenticular and spiral galaxies with features, such as spiral arms and a distinct halo . At the core, most galaxies have

473-401: Is assumed that mean anomaly is zero at the epoch (by choosing the appropriate definition of the epoch), leaving only the five other orbital elements to be specified. Different sets of elements are used for various astronomical bodies. The eccentricity, e , and either the semi-major axis, a , or the distance of periapsis, q , are used to specify the shape and size of an orbit. The longitude of

SECTION 10

#1732848459520

516-412: Is because the problem contains six degrees of freedom . These correspond to the three spatial dimensions which define position ( x , y , z in a Cartesian coordinate system ), plus the velocity in each of these dimensions. These can be described as orbital state vectors , but this is often an inconvenient way to represent an orbit, which is why Keplerian elements are commonly used instead. Sometimes

559-584: Is classified by the IAU as a small Solar System body (SSSB). These come in many non-spherical shapes which are lumpy masses accreted haphazardly by in-falling dust and rock; not enough mass falls in to generate the heat needed to complete the rounding. Some SSSBs are just collections of relatively small rocks that are weakly held next to each other by gravity but are not actually fused into a single big bedrock . Some larger SSSBs are nearly round but have not reached hydrostatic equilibrium. The small Solar System body 4 Vesta

602-436: Is greater than one, the trajectory is a hyperbola . If the eccentricity is equal to one, the trajectory is a parabola . Regardless of eccentricity, the orbit degenerates to a radial trajectory if the angular momentum equals zero. Given an inertial frame of reference and an arbitrary epoch (a specified point in time), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit. This

645-524: Is large enough to have undergone at least partial planetary differentiation. Stars like the Sun are also spheroidal due to gravity's effects on their plasma , which is a free-flowing fluid . Ongoing stellar fusion is a much greater source of heat for stars compared to the initial heat released during their formation. The table below lists the general categories of bodies and objects by their location or structure. Orbital elements Orbital elements are

688-457: Is shown as the red angle ν in the diagram, and the mean anomaly is not shown. The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system. Note that non-elliptic trajectories also exist, but are not closed, and are thus not orbits. If the eccentricity

731-561: Is the NASA / NORAD "two-line elements" (TLE) format, originally designed for use with 80 column punched cards, but still in use because it is the most common format, and 80-character ASCII records can be handled efficiently by modern databases. Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable. Orbital positions can be calculated from TLEs through simplified perturbation models ( SGP4 / SDP4 / SGP8 / SDP8). Example of

774-571: The action variables and are more elaborate combinations of the Keplerian elements a , e , and i . Delaunay variables are used to simplify perturbative calculations in celestial mechanics, for example while investigating the Kozai–Lidov oscillations in hierarchical triple systems. The advantage of the Delaunay variables is that they remain well defined and non-singular (except for h , which can be tolerated) when e and / or i are very small: When

817-507: The Andromeda nebula as a different galaxy, along with many others far from the Milky Way. The universe can be viewed as having a hierarchical structure. At the largest scales, the fundamental component of assembly is the galaxy . Galaxies are organized into groups and clusters , often within larger superclusters , that are strung along great filaments between nearly empty voids , forming

860-498: The Sun located in the center of the Solar System . Johannes Kepler discovered Kepler's laws of planetary motion , which are properties of the orbits that the astronomical bodies shared; this was used to improve the heliocentric model. In 1584, Giordano Bruno proposed that all distant stars are their own suns, being the first in centuries to suggest this idea. Galileo Galilei was one of

903-470: The parameters required to uniquely identify a specific orbit . In celestial mechanics these elements are considered in two-body systems using a Kepler orbit . There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics . A real orbit and its elements change over time due to gravitational perturbations by other objects and

SECTION 20

#1732848459520

946-484: The photoelectric photometer allowed astronomers to accurately measure the color and luminosity of stars, which allowed them to predict their temperature and mass. In 1913, the Hertzsprung-Russell diagram was developed by astronomers Ejnar Hertzsprung and Henry Norris Russell independently of each other, which plotted stars based on their luminosity and color and allowed astronomers to easily examine stars. It

989-491: The protoplanetary disks that surround newly formed stars. The various distinctive types of stars are shown by the Hertzsprung–Russell diagram (H–R diagram)—a plot of absolute stellar luminosity versus surface temperature. Each star follows an evolutionary track across this diagram. If this track takes the star through a region containing an intrinsic variable type, then its physical properties can cause it to become

1032-3674: The 3 coordinates in the I-J-K system given the 3 (or 2) coordinates in the x-y-z system, is represented by the inverse matrix. According to the rules of matrix algebra , the inverse matrix of the product of the 3 rotation matrices is obtained by inverting the order of the three matrices and switching the signs of the three Euler angles. That is, [ i 1 i 2 i 3 j 1 j 2 j 3 k 1 k 2 k 3 ] = [ cos ⁡ Ω − sin ⁡ Ω 0 sin ⁡ Ω cos ⁡ Ω 0 0 0 1 ] [ 1 0 0 0 cos ⁡ i − sin ⁡ i 0 sin ⁡ i cos ⁡ i ] [ cos ⁡ ω − sin ⁡ ω 0 sin ⁡ ω cos ⁡ ω 0 0 0 1 ] ; {\displaystyle {\begin{bmatrix}i_{1}&i_{2}&i_{3}\\j_{1}&j_{2}&j_{3}\\k_{1}&k_{2}&k_{3}\end{bmatrix}}={\begin{bmatrix}\cos \Omega &-\sin \Omega &0\\\sin \Omega &\cos \Omega &0\\0&0&1\end{bmatrix}}\,{\begin{bmatrix}1&0&0\\0&\cos i&-\sin i\\0&\sin i&\cos i\end{bmatrix}}\,{\begin{bmatrix}\cos \omega &-\sin \omega &0\\\sin \omega &\cos \omega &0\\0&0&1\end{bmatrix}}\,;} where I ^ = i 1 x ^ + i 2 y ^ + i 3 z ^   ; J ^ = j 1 x ^ + j 2 y ^ + j 3 z ^   ; K ^ = k 1 x ^ + k 2 y ^ + k 3 z ^   . {\displaystyle {\begin{aligned}\mathbf {\hat {I}} &=i_{1}\mathbf {\hat {x}} +i_{2}\mathbf {\hat {y}} +i_{3}\mathbf {\hat {z}} ~;\\\mathbf {\hat {J}} &=j_{1}\mathbf {\hat {x}} +j_{2}\mathbf {\hat {y}} +j_{3}\mathbf {\hat {z}} ~;\\\mathbf {\hat {K}} &=k_{1}\mathbf {\hat {x}} +k_{2}\mathbf {\hat {y}} +k_{3}\mathbf {\hat {z}} ~.\\\end{aligned}}} The transformation from x̂ , ŷ , ẑ to Euler angles Ω , i , ω is: Ω = arg ⁡ ( − z 2 , z 1 ) i = arg ⁡ ( z 3 , z 1 2 + z 2 2 ) ω = arg ⁡ ( y 3 , x 3 ) {\displaystyle {\begin{aligned}\Omega &=\operatorname {arg} \left(-z_{2},z_{1}\right)\\i&=\operatorname {arg} \left(z_{3},{\sqrt {{z_{1}}^{2}+{z_{2}}^{2}}}\right)\\\omega &=\operatorname {arg} \left(y_{3},x_{3}\right)\\\end{aligned}}} where arg( x , y ) signifies

1075-525: The Keplerian elements define an ellipse , parabola , or hyperbola . Real orbits have perturbations, so a given set of Keplerian elements accurately describes an orbit only at the epoch. Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, the nonsphericity of the primary, atmospheric drag , relativistic effects , radiation pressure , electromagnetic forces , and so on. Keplerian elements can often be used to produce useful predictions at times near

1118-400: The ascending node, Ω , the inclination, i , and the argument of periapsis, ω , or the longitude of periapsis, ϖ , specify the orientation of the orbit in its plane. Either the longitude at epoch, L 0 , the mean anomaly at epoch, M 0 , or the time of perihelion passage, T 0 , are used to specify a known point in the orbit. The choices made depend whether the vernal equinox or

1161-444: The effects of general relativity . A Kepler orbit is an idealized, mathematical approximation of the orbit at a particular time. The traditional orbital elements are the six Keplerian elements , after Johannes Kepler and his laws of planetary motion . When viewed from an inertial frame , two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass . When viewed from

1204-524: The epoch is considered a "seventh" orbital parameter, rather than part of the reference frame. If the epoch is defined to be at the moment when one of the elements is zero, the number of unspecified elements is reduced to five. (The sixth parameter is still necessary to define the orbit; it is merely numerically set to zero by convention or "moved" into the definition of the epoch with respect to real-world clock time.) Keplerian elements can be obtained from orbital state vectors (a three-dimensional vector for

1247-439: The epoch. Alternatively, real trajectories can be modeled as a sequence of Keplerian orbits that osculate ("kiss" or touch) the real trajectory. They can also be described by the so-called planetary equations , differential equations which come in different forms developed by Lagrange , Gauss , Delaunay , Poincaré , or Hill . Keplerian elements parameters can be encoded as text in a number of formats. The most common of them

1290-475: The first astronomers to use telescopes to observe the sky, in 1610 he observed the four largest moons of Jupiter , now named the Galilean moons . Galileo also made observations of the phases of Venus , craters on the Moon , and sunspots on the Sun. Astronomer Edmond Halley was able to successfully predict the return of Halley's Comet , which now bears his name, in 1758. In 1781, Sir William Herschel discovered

1333-521: The human eye were discovered, and new telescopes were made that made it possible to see astronomical objects in other wavelengths of light. Joseph von Fraunhofer and Angelo Secchi pioneered the field of spectroscopy , which allowed them to observe the composition of stars and nebulae, and many astronomers were able to determine the masses of binary stars based on their orbital elements . Computers began to be used to observe and study massive amounts of astronomical data on stars, and new technologies such as

Luminary - Misplaced Pages Continue

1376-421: The mean motion ( n ) into the polynomial as one of the coefficients. The appearance will be that L or M are expressed in a more complicated manner, but we will appear to need one fewer orbital element. Mean motion can also be obscured behind citations of the orbital period P . The angles Ω , i , ω are the Euler angles (corresponding to α , β , γ in the notation used in that article) characterizing

1419-518: The movements of the bodies very important as they used these objects to help navigate over long distances, tell between the seasons, and to determine when to plant crops. During the Middle-Ages , cultures began to study the movements of these bodies more closely. Several astronomers of the Middle-East began to make detailed descriptions of stars and nebulae, and would make more accurate calendars based on

1462-546: The movements of these stars and planets. In Europe , astronomers focused more on devices to help study the celestial objects and creating textbooks, guides, and universities to teach people more about astronomy. During the Scientific Revolution , in 1543, Nicolaus Copernicus's heliocentric model was published. This model described the Earth , along with all of the other planets as being astronomical bodies which orbited

1505-499: The new planet Uranus , being the first discovered planet not visible by the naked eye. In the 19th and 20th century, new technologies and scientific innovations allowed scientists to greatly expand their understanding of astronomy and astronomical objects. Larger telescopes and observatories began to be built and scientists began to print images of the Moon and other celestial bodies on photographic plates. New wavelengths of light unseen by

1548-399: The node are used as the primary reference. The semi-major axis is known if the mean motion and the gravitational mass are known. It is also quite common to see either the mean anomaly ( M ) or the mean longitude ( L ) expressed directly, without either M 0 or L 0 as intermediary steps, as a polynomial function with respect to time. This method of expression will consolidate

1591-5052: The orientation of the coordinate system where: Then, the transformation from the Î , Ĵ , K̂ coordinate frame to the x̂ , ŷ , ẑ frame with the Euler angles Ω , i , ω is: x 1 = cos ⁡ Ω ⋅ cos ⁡ ω − sin ⁡ Ω ⋅ cos ⁡ i ⋅ sin ⁡ ω   ; x 2 = sin ⁡ Ω ⋅ cos ⁡ ω + cos ⁡ Ω ⋅ cos ⁡ i ⋅ sin ⁡ ω   ; x 3 = sin ⁡ i ⋅ sin ⁡ ω ; y 1 = − cos ⁡ Ω ⋅ sin ⁡ ω − sin ⁡ Ω ⋅ cos ⁡ i ⋅ cos ⁡ ω   ; y 2 = − sin ⁡ Ω ⋅ sin ⁡ ω + cos ⁡ Ω ⋅ cos ⁡ i ⋅ cos ⁡ ω   ; y 3 = sin ⁡ i ⋅ cos ⁡ ω   ; z 1 = sin ⁡ i ⋅ sin ⁡ Ω   ; z 2 = − sin ⁡ i ⋅ cos ⁡ Ω   ; z 3 = cos ⁡ i   ; {\displaystyle {\begin{aligned}x_{1}&=\cos \Omega \cdot \cos \omega -\sin \Omega \cdot \cos i\cdot \sin \omega \ ;\\x_{2}&=\sin \Omega \cdot \cos \omega +\cos \Omega \cdot \cos i\cdot \sin \omega \ ;\\x_{3}&=\sin i\cdot \sin \omega ;\\\,\\y_{1}&=-\cos \Omega \cdot \sin \omega -\sin \Omega \cdot \cos i\cdot \cos \omega \ ;\\y_{2}&=-\sin \Omega \cdot \sin \omega +\cos \Omega \cdot \cos i\cdot \cos \omega \ ;\\y_{3}&=\sin i\cdot \cos \omega \ ;\\\,\\z_{1}&=\sin i\cdot \sin \Omega \ ;\\z_{2}&=-\sin i\cdot \cos \Omega \ ;\\z_{3}&=\cos i\ ;\\\end{aligned}}} [ x 1 x 2 x 3 y 1 y 2 y 3 z 1 z 2 z 3 ] = [ cos ⁡ ω sin ⁡ ω 0 − sin ⁡ ω cos ⁡ ω 0 0 0 1 ] [ 1 0 0 0 cos ⁡ i sin ⁡ i 0 − sin ⁡ i cos ⁡ i ] [ cos ⁡ Ω sin ⁡ Ω 0 − sin ⁡ Ω cos ⁡ Ω 0 0 0 1 ] ; {\displaystyle {\begin{bmatrix}x_{1}&x_{2}&x_{3}\\y_{1}&y_{2}&y_{3}\\z_{1}&z_{2}&z_{3}\end{bmatrix}}={\begin{bmatrix}\cos \omega &\sin \omega &0\\-\sin \omega &\cos \omega &0\\0&0&1\end{bmatrix}}\,{\begin{bmatrix}1&0&0\\0&\cos i&\sin i\\0&-\sin i&\cos i\end{bmatrix}}\,{\begin{bmatrix}\cos \Omega &\sin \Omega &0\\-\sin \Omega &\cos \Omega &0\\0&0&1\end{bmatrix}}\,;} where x ^ = x 1 I ^ + x 2 J ^ + x 3 K ^   ; y ^ = y 1 I ^ + y 2 J ^ + y 3 K ^   ; z ^ = z 1 I ^ + z 2 J ^ + z 3 K ^   . {\displaystyle {\begin{aligned}\mathbf {\hat {x}} &=x_{1}\mathbf {\hat {I}} +x_{2}\mathbf {\hat {J}} +x_{3}\mathbf {\hat {K}} ~;\\\mathbf {\hat {y}} &=y_{1}\mathbf {\hat {I}} +y_{2}\mathbf {\hat {J}} +y_{3}\mathbf {\hat {K}} ~;\\\mathbf {\hat {z}} &=z_{1}\mathbf {\hat {I}} +z_{2}\mathbf {\hat {J}} +z_{3}\mathbf {\hat {K}} ~.\\\end{aligned}}} The inverse transformation, which computes

1634-416: The other provided the standard gravitational parameter , GM , is given for the central body. Instead of the mean anomaly at epoch , the mean anomaly M , mean longitude , true anomaly ν 0 , or (rarely) the eccentric anomaly might be used. Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time t must be specified as a seventh orbital element. Sometimes it

1677-415: The polar argument that can be computed with the standard function atan2(y,x) available in many programming languages. Under ideal conditions of a perfectly spherical central body, zero perturbations and negligible relativistic effects, all orbital elements except the mean anomaly are constants. The mean anomaly changes linearly with time, scaled by the mean motion , n = μ

1720-471: The position and another for the velocity) by manual transformations or with computer software. Other orbital parameters can be computed from the Keplerian elements such as the period , apoapsis, and periapsis . (When orbiting the Earth, the last two terms are known as the apogee and perigee.) It is common to specify the period instead of the semi-major axis a in Keplerian element sets, as each can be computed from

1763-417: The secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary. Two elements define the shape and size of the ellipse: Two elements define the orientation of the orbital plane in which the ellipse is embedded: The remaining two elements are as follows: The mean anomaly M is a mathematically convenient fictitious "angle" which does not correspond to

Luminary - Misplaced Pages Continue

1806-399: The various condensing nebulae. The great variety of stellar forms are determined almost entirely by the mass, composition and evolutionary state of these stars. Stars may be found in multi-star systems that orbit about each other in a hierarchical organization. A planetary system and various minor objects such as asteroids, comets and debris, can form in a hierarchical process of accretion from

1849-572: Was found that stars commonly fell on a band of stars called the main-sequence stars on the diagram. A refined scheme for stellar classification was published in 1943 by William Wilson Morgan and Philip Childs Keenan based on the Hertzsprung-Russel Diagram. Astronomers also began debating whether other galaxies existed beyond the Milky Way , these debates ended when Edwin Hubble identified

#519480