Steinhaus theorem

2013-04-21T12:35:57Z

92.23.177.43: /* Statement */ why Borel?

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{{hatnote|Other templates redirecting here: '''{{t|Planet}}, {{t|Infobox minor planet}}, {{t|Infobox Planet}}, {{t|Infobox Moon}}, {{t|Infobox Nonstellar body}}'''}}
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This template has been designed for the presentation of information on non-stellar astronomical bodies: [[planet]]s (including [[extrasolar planet]]s), [[dwarf planet]]s, [[moon]]s and [[minor planet]]s. Some parameters will not be applicable to individual types; these may be omitted without any problems to the template's function.

== Usage notes ==
* This template expects that the {{tag|references|single}} tag will be present in articles setting the ''minorplanet'' parameter to ''yes''. Pages without a {{tag|references|single}} tag will show {{color|red|Cite error: There are {{tag|ref|open}} tags on this page for a group named "note", but the references will not show without a {{tag|ref group="note"|open}} tag; see Help:Cite errors.}} at the bottom of the page.
* When parameters relating to [[minor planet]]s are used (e.g. ''mp_name''), several subheadings and wikilinks within the template are made specific to minor planets. For standard formulas for computable values and sources of physical data, see [[Template talk:Infobox Nonstellar body#Computed values]] and [[Standard asteroid physical characteristics]].
* The Proper Orbital Element "Proper orbital period" (in Julian Years and Days) is calculated within the template from Proper Mean Motion and so does not require a parameter.
* For bodies orbiting bodies other than the [[Sun]], include the parameter <tt>|apsis = ''(appropriate suffix)''</tt>. This will replace various parameters' default "-helion" suffix to the suffix set by the ''[[apsis]]'' parameter. For example, setting <tt>|apsis = astron</tt> converts the labels "perihelion", "aphelion" and "Argument of perihelion" into "periastron", "apastron" and "Argument of periastron" respectively.

== Recommended parameters ==

For a list of all parameters, see [[#All parameters|All parameters]] below.

=== Exosolar planets ===

{{Infobox planet
| name = ''name''
| image = image
| caption = ''caption''
| apsis = astron
| discoverer = ''discoverer''
| discovery_site = ''discovery_site''
| discovered = ''discovered''
| discovery_method = ''discovery_method''
| alt_names = ''alt_names''
| periastron = ''periastron''
| apoastron = ''apoastron''
| semimajor = ''semimajor''
| avg_speed = ''avg_speed''
| eccentricity = ''eccentricity''
| period = ''period''
| inclination = ''inclination''
| angular_dist = ''angular_dist''
| long_periastron = ''long_periastron''
| time_periastron = ''time_periastron''
| semi-amplitude = ''semi-amplitude''
| mean_radius = ''mean_radius''
| surface_area = ''surface_area''
| volume = ''volume''
| density = ''density''
| surface_grav = ''surface_grav''
| escape_velocity = ''escape_velocity''
| mass = ''mass''
| albedo = ''albedo''
| single_temperature = ''single_temperature''
}}
<pre style="font-size: 95%; overflow: auto;">
{{Infobox planet
| name = 
| image =
| caption =
| apsis = astron
| discovery_ref = 
| discoverer =
| discovery_site =
| discovered = 
| discovery_method =
| alt_names = 
| periastron =
| apoastron =
| semimajor =
| avg_speed = 
| eccentricity =
| period = 
| inclination =
| angular_dist =
| long_periastron = 
| time_periastron = 
| semi-amplitude =
| mean_radius =
| surface_area =
| volume =
| density =
| mass =
| surface_grav = 
| escape_velocity =
| albedo =
| single_temperature =
}}
</pre>
{{clear}}

=== Minor planets ===

{{Infobox planet
| width =
| name = ''name''
| symbol = ''symbol''
| image = image
| caption = ''caption''
| discovery_ref = <ref>''discovery_ref''</ref>
| discoverer = ''discoverer''
| discovered = ''discovered''
| discovery_site = ''discovery site''
| mp_name = ''mp_name''
| mp_category = ''mp_category''
| alt_names = ''alt_names''
| orbit_ref = <ref>''orbit_ref''</ref>
| epoch = ''epoch''
| aphelion = ''aphelion''
| perihelion = ''perihelion''
| semimajor = ''semimajor''
| eccentricity = ''eccentricity''
| period = ''period''
| avg_speed = ''avg_speed''
| inclination = ''inclination''
| mean_anomaly = ''mean_anomaly''
| asc_node = ''asc_node''
| arg_peri = ''arg_peri''
| p_orbit_ref = <ref>''p_orbit_ref''</ref>
| p_semimajor = ''p_semi_major''
| p_eccentricity = ''p_eccentricity''
| p_inclination = ''p_inclination''
| p_mean_motion = 360
| perihelion_rate = ''perihelion_rate''
| node_rate = ''node_rate''
| satellites = ''satellites''
| dimensions = ''dimensions''
| mass = ''mass''
| density = ''density''
| surface_grav = ''surface_grav''
| escape_velocity = ''escape_velocity''
| sidereal_day = ''sidereal_day''
| axial_tilt = ''axial_tilt''
| pole_ecliptic_lat = ''pole_ecliptic_lat''
| pole_ecliptic_lon = ''pole_ecliptic_lon''
| albedo = ''albedo''
| temp_name1 = [[Kelvin]]
| mean_temp_1 = 150
| max_temp_1 = 200
| temp_name2 = Celsius
| max_temp_2 = {{val|-73}}
| spectral_type = ''spectral_type''
| abs_magnitude = ''abs_magnitude''
}}
<pre style="font-size: 95%; overflow: auto;">
{{Infobox planet
| width = (default is 22em)
| name =
| symbol =
| image =
| caption =
| discovery_ref = <ref>....</ref>
| discoverer =
| discovered =
| discovery_site = ''discovery_site''
| mp_name = 
| mp_category = 
| alt_names = 
| orbit_ref = <ref>Osculating Orbital Elements</ref>
| epoch =
| aphelion =
| perihelion =
| semimajor =
| eccentricity =
| period =
| avg_speed = 
| inclination =
| asc_node = 
| mean_anomaly =
| arg_peri = 
| p_orbit_ref = <ref>Proper Orbital Elements</ref>
| p_semimajor =
| p_eccentricity =
| p_inclination =
| p_mean_motion = 
| perihelion_rate = 
| node_rate = 
| satellites =
| dimensions =
| mass =
| density =
| surface_grav = 
| escape_velocity =
| sidereal_day =
| axial_tilt =
| pole_ecliptic_lat = 
| pole_ecliptic_lon = 
| albedo =
| temp_name1 =
| mean_temp_1 =
| max_temp_1 =
| temp_name2 =
| max_temp_2 =
| spectral_type =
| abs_magnitude = 
}}
</pre>

See [[Wikipedia:WikiProject Astronomical objects/Infoboxes#Minor planets (asteroids, comets, Kuiperoids, etc.)|here]] for  means of semi-automating this template's transclusion as regards minor planets.
* For satellites of minor planets, include the parameter <tt>|apsis = astron</tt> (see [[#Usage notes|above]]).
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== Notes on usage ==

: ''From [[Wikipedia:WikiProject Astronomical objects/Infoboxes]]''

Most of these entries should be measured in [[SI]] units. Some of them, however, should have more "human-accessible" units, in addition to SI units: several such cases are indicated with a second unit name in brackets. In the case of times (orbital periods, rotation), it is best to give all periods in days for comparison purposes, and provide a translation (in parentheses) into years, days, hours, etc.; whatever is most appropriate for the duration being described.

This template is very flexible. Moons with no atmosphere whatsoever could skip the atmospheric composition section entirely, for example (though atmospheric density would still be listed). Moons also wouldn't have their orbital radii listed in AU, since AUs are such large units. For planets, use "perihelion" and "aphelion" instead of "periapsis" and "apoapsis."

In the case of "number of moons" and "is a moon of", only one of these rows will be used by any given object. There aren't any moons with moons (yet), though perhaps "co-orbital with" might be a useful row to add in a few cases.

'''On orbital characteristics:''' The orbital circumference should be computed from the semi-major axis using [[Ellipse#Circumference|Ramanujan's approximation]] for ellipses. The ratio of that circumference to the period then gives the average orbital speed. The minimum and maximum speeds follow from Kepler's laws: <math>v_{max} = 2\pi a^2 \frac{\sqrt{1-e^2}}{T a (1-e)}</math> and <math>v_{min} = 2\pi a^2 \frac{\sqrt{1-e^2}}{T a (1+e)}</math>. Note that, by convention, all orbital parameters are given in the primocentric reference system (heliocentric for the planets).

'''On proper orbital elements:''' The formulae used by the template to convert from proper mean motion to proper orbital motion are: Oyears = 360 / M   and Odays = 365.25 x Oyears

'''On physical characteristics:''' The surface area and volume of non-spherical objects (e.g. moonlets, asteroids) must use the proper [[ellipsoid]] formulae, because even slight departures from sphericity will make a large difference, particularly for the area.

'''On the subject of obliquity:''' [[Obliquity]] is the angle between the object's axis of rotation and the normal to the plane of its orbit. Do not confuse this with the Tilt listed in the JPL pages, which is a measure of the angle between the local [[Laplace plane]] and the primary's equatorial plane. In fact, most inner moons have [[synchronous rotation]]s, so their obliquities will be, by definition, zero. Outer moons simply have not been seen from close up enough to determine their true obliquities (although [[Phoebe (moon)|Phoebe]], recently seen by the [[Cassini probe]], is an exception; see [[Talk:Phoebe (moon)]] for the derivation of its obliquity).

=== All parameters ===

<pre style="font-size: 95%; overflow: auto;">
{{Infobox planet
| width = 
| name =
| symbol = 
| background =
| image = 
| caption =
| apsis =
| discovery_ref = <ref>...</ref>
| discoverer =
| discovery_site =
| discovered =
| discovery_method =
| mp_name =
| pronounced =
| named_after =
| mp_category =
| alt_names =
| orbit_ref = <ref>Osculating Orbital Elements</ref>
| orbit_diagram = 
| epoch =
| aphelion =
| perihelion =
| periastron =
| apoastron =
| periapsis =
| apoapsis =
| semimajor =
| mean_orbit_radius =
| eccentricity =
| period =
| synodic_period =
| avg_speed = 
| mean_anomaly =
| inclination =
| angular_dist = 
| asc_node = 
| long_periastron = 
| time_periastron = 
| arg_peri = 
| semi-amplitude =
| p_orbit_ref = <ref>Proper Orbital Elements</ref>
| p_semimajor =
| p_eccentricity =
| p_inclination =
| p_mean_motion = 
| perihelion_rate = 
| node_rate = 
| satellite_of =
| satellites =
| dimensions =
| flattening =
| equatorial_radius =
| polar_radius =
| mean_radius =
| circumference =
| surface_area =
| volume =
| mass =
| density =
| surface_grav = 
| escape_velocity =
| sidereal_day =
| rot_velocity = 
| axial_tilt =
| right_asc_north_pole = 
| declination = 
| pole_ecliptic_lat = 
| pole_ecliptic_lon = 
| albedo =
| single_temperature =
| temp_name1 =
| min_temp_1 =
| mean_temp_1 =
| max_temp_1 =
| temp_name2 =
| min_temp_2 =
| mean_temp_2 =
| max_temp_2 =
| spectral_type =
| magnitude = 
| abs_magnitude = 
| angular_size =
| pronounced = {{IPA-en||}}
| adjectives =
| atmosphere_ref = <ref>...</ref>
| surface_pressure =
| scale_height =
| atmosphere_composition =
}}
</pre>

{{clear}}

== Computed values ==

This section documents how some minor planet parameters may be computed when they're not directly measured. More detail can be found in [[Standard asteroid physical characteristics]]. 

=== Average orbital speed ===
This is very simply the orbital circumference divided by the orbital period.

The exact [[circumference]] of an ellipse is <math>4 a E(e)</math>,
where ''a'' is the [[semi-major axis]], ''e'' the [[eccentricity (mathematics)|eccentricity]], and the function ''E'' is the complete [[elliptic integral]] of the [[Elliptic integral#Complete elliptic integral of the second kind|second kind]]. This gives
: <math>
v_o = \frac{4 a E(e)}{T}
</math>
''E'' is close to <math> \pi / 2</math> when ''e'' is small.

An approximation using a [[taylor series]] expansion is
: <math>
v_o = \frac{2\pi a}{T}\left[1-\frac{e^2}{4}-\frac{3e^4}{64} - \dots \right]
</math>

[[Wikipedia:WikiProject Astronomical objects/Infoboxes#Minor planets (asteroids, comets, Kuiperoids, etc.)|AstOrb Browser]] computes a velocity using [[Srinivasa Ramanujan|Ramanujan]]'s approximation for an [[Ellipse#Circumference|ellipse's circumference]]:
: <math>v_o \approx \frac{\pi}{T} \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right] </math>
Where ''b'' is the orbit's [[semi-minor axis]]:
: <math>b = a \sqrt{1-e^2}\,\!</math>

=== Surface gravity ===

{{main|Surface gravity}}
For a ''spherical'' body of mass ''m'', and radius ''r'', the [[gravitational acceleration]] at the surface, is given by

: <math>g_{\rm spherical} = \frac{GM}{r^2}\,\!</math>

Where ''G'' = 6.6742{{e|−11}} m3s−2kg−1 is the [[Gravitational constant]], ''M'' is the mass of the body, and ''r'' its radius. This value is very approximate, as most minor planets are far from spherical.

For ''irregularly shaped bodies'', the surface gravity will differ appreciably with location. However, at the outermost point/s, where the distance to the [[centre of mass]] is the greatest, the surface gravity is still given by a simple formula, a slightly modified version of the above that uses the largest radius <math>r_{\rm max}</math>

: <math>g_{\rm outer} = \frac{GM}{r_{\rm max}^2}\,.\!</math>

because all the body's mass is contained within this radius.

On a ''rotating body'', the apparent weight experienced by an object on the surface is reduced by the [[centrifugal force]], when one is away from the poles. The centrifugal acceleration experienced at the equator is

: <math>g_{\rm centrifugal} = -\left(\frac{2\pi}{T}\right)^2 r_{\rm eq}</math>

where ''T'' is the rotation period in seconds, and <math>r_{\rm eq}</math> is the equatorial radius (usually also the maximum radius used above). The negative sign indicates that it acts in the opposite direction to the gravitational acceleration ''g''.

The ''effective surface gravity at the equator'' is then

: <math> g_{\rm effective} \approx g_{\rm gravitational} + g_{\rm centrifugal}
= g_{\rm gravitational} - |g_{\rm centrifugal}|\ .</math>

=== Escape velocity ===

For surface gravity ''g'' and radius ''r'', the escape velocity is:
: <math>v_e = \sqrt{2gr}</math>

This value is much less sensitive to the factors affecting the surface gravity, mentioned above.

=== Temperature ===

For asteroid albedo ''α'', semimajor axis ''a'', [[solar luminosity]] ''<math>L_0</math>'', and asteroid [[infrared]] emissivity ''ε'' (usually taken to be ~0.9), the approximate mean temperature ''T'' is given by:
: <math>T = \left ( \frac{(1 - \alpha) L_0}{\epsilon \sigma 16 \pi a^2} \right )^{\frac{1}{4}}</math>

Where ''σ'' is [[Stefan-Boltzmann constant]]. See also {{cite book
| author = Torrence V. Johnson, Paul R. Weissman, Lucy-Ann A. McFadden
| year = 2007 | title = Encyclopedia of the Solar System
| pages = 294 | publisher = Elsevier
| isbn = 978-0-12-088589-3 }}.

== Microformat ==

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== References ==

{{reflist}}
<includeonly>

[[Category:Astronomy infobox templates|Planet]]


</includeonly>

<includeonly>
[[ba:Ҡалып:Infobox күк есеме]]
[[or:Infobox planet/doc]]
</includeonly>

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Steinhaus theorem