|
|
| Line 1: |
Line 1: |
| {{For|specific applications of Kepler's equation|Kepler's laws of planetary motion}}
| | Decoration Books is one particular of the finest guides that assist you to decorate your preferred locations and things. In simple words with the assist of decorating book you can decorate your house, garden, office, cakes, cookies, etc. It has the greatest collection in most of the topics that a decorator seeks for. With decorating book you can make a nicely-decorated output. It does not matters whether or not you are a excellent decorator or not but with an suitable decorating book you can decorate simply and quickly.<br><br> |
| [[File:Kepler's equation solutions.PNG|thumb|right|Kepler's equation solutions for five different eccentricities between 0 and 1]]
| |
| In [[orbital mechanics]], '''Kepler's equation''' relates various geometric properties of the orbit of a body subject to a [[central force]]. | |
|
| |
|
| It was first derived by [[Johannes Kepler]] in 1609 in Chapter 60 of his ''[[Astronomia nova]]'',<ref>{{cite book |last=Kepler |first=Johannes |year=1609 |title=[[Astronomia Nova|Astronomia Nova Aitiologētos, Seu Physica Coelestis, tradita commentariis De Motibus Stellæ Martis, Ex observationibus G. V. Tychonis Brahe]] |chapter=LX. Methodus, ex hac Physica, hoc est genuina & verissima hypothesi, extruendi utramque partem æquationis, & distantias genuinas: quorum utrumque simul per vicariam fieri hactenus non potuit. argumentum falsæ hypotheseos|language=Latin|pages=299–300|url=http://www.e-rara.ch/zut/content/pageview/162861}}</ref><ref>{{cite book |last=Aaboe |first=Asger |year=2001 |title=Episodes from the Early History of Astronomy |publisher=Springer|pages=146–147|url=http://books.google.com/books?id=yK8Tp0izorMC&pg=PA146|isbn=9780387951362}}</ref> and in book V of his ''Epitome of Copernican Astronomy'' (1621) Kepler proposed an iterative solution to the equation.<ref>{{cite book |last=Kepler |first=Johannes |title=Epitome astronomiæ Copernicanæ usitatâ formâ Quæstionum & Responsionum conscripta, inq; VII. Libros digesta, quorum tres hi priores sunt de Doctrina Sphæricâ |year=1621|chapter=Libri V. Pars altera. |language=Latin|pages=695–696|url=http://www.e-rara.ch/zut/content/pageview/956468}}</ref><ref>{{cite journal|url=http://adsabs.harvard.edu/full/2000JHA....31..339S |title=Kepler's Iterative Solution to Kepler's Equation |first=N. M. |last=Swerdlow|journal=[[Journal for the History of Astronomy]] |volume=31 |pages=339–341 |year=2000 |bibcode=2000JHA....31..339S}}</ref> The equation has played an important role in the history of both physics and mathematics, particularly classical [[celestial mechanics]].
| | Decoration Books: Benefits<br><br>Decorating book includes skilled decorating ideas, clear directions, lavish photographs and effortless tasks that aid you to decorate with ease. Decoration Books help you to make uniquely decorated house.<br><br>There are distinct varieties of decoration books available in the marketplace such as cookie decorating book, interior decorating book and so on. A single of such varieties of painting book is Cake decorating book.<br><br>Cake Decorating Book is a ideal decision to decorate the cake. This book offers you important tips, together with illustrated pictures to decorate your birthday cakes, Christmas cake, wedding cake and celebration cake. My uncle learned about [http://scriptogr.am/sourcereviewsswitch book a celebrity] by browsing Google Books. If you want to decorate the cake for any special occasion then you require to buy a cake decorating book.<br><br>Now days you can locate enormous collection of such books on the web. This book can help you to decorate your cake in a expert manner. Cake Decorating Books not only assist you to decorate your cake but it also aids you to make your cake scrumptious, which is an added benefit. Cake decorating book, which is supplied on the web, requires no basic encounter. With the aid of cake decorating book you can prepare exclusive decorated pastries or cakes. This book is published in such a way that no one can refrain oneself from creating a decorated cake. Every chapter carries new tips and suggestions.<br><br>It is essential that one particular must decorate the cake as it gives a distinct and attractive appear that produces the desire to eat it. This cake decorating book is also ideal for novices, which provides step by step instructions and guidance.. Be taught additional resources on our affiliated link - Click here: [http://about.me/guidecelebrityxdt details]. For extra information, consider checking out: [http://www.cultureinside.com/123/section.aspx/Member/celebritysourcedgk/ like i said].<br><br>If you're ready to see more in regards to [http://www.purevolume.com/listeners/ossifieddonor5075 cobra health insurance] review our own web site. |
| | |
| ==Equation==
| |
| | |
| '''Kepler's equation''' is
| |
| | |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math> M = E -\varepsilon \sin E </math>
| |
| |border colour = #50C878
| |
| |background colour = #ECFCF4}}
| |
| | |
| where ''M'' is the [[mean anomaly]], ''E'' is the [[eccentric anomaly]], and ε is the [[Eccentricity (mathematics)|eccentricity]]. Kepler's equation is a [[transcendental equation]] because [[sine]] is a [[transcendental function]], meaning it cannot be solved for ''E'' [[algebraic function|algebraically]]. [[Numerical analysis]] and [[Series (mathematics)|series]] expansions are generally required to evaluate ''E''.
| |
| | |
| == Alternate forms ==
| |
| | |
| There are several forms of Kepler's equation. Each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits (0 ≤ ε < 1). The hyperbolic Kepler equation is used for hyperbolic orbits (ε ≫ 1). The radial Kepler equation is used for linear (radial) orbits (ε = 1). [[Parabolic trajectory#Barker's equation|Barker's equation]] is used for parabolic orbits (ε = 1). When ε = 1, Kepler's equation is not associated with an orbit.
| |
| | |
| When ε = 0, the orbit is circular. Increasing ε causes the circle to flatten into an ellipse. When ε = 1, the orbit is completely flat, and it appears to be a either a segment if the orbit is closed, or a ray if the orbit is open. An infinitesimal increase to ε results in a hyperbolic orbit with a turning angle of 180 degrees, and the orbit appears to be a ray. Further increases reduce the turning angle, and as ε goes to infinity, the orbit becomes a straight line of infinite length.
| |
| | |
| ===Hyperbolic Kepler equation===
| |
| | |
| The Hyperbolic Kepler equation is:
| |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math> M = \varepsilon \sinh H - H </math>
| |
| |border colour|background colour }}
| |
| | |
| where ''H'' is the hyperbolic eccentric anomaly.
| |
| This equation is derived by multiplying Kepler's equation by the [[Imaginary number|square root of −1]]; ''i'' = √(−1) for [[imaginary unit]], and replacing | |
| | |
| :<math> E = i H </math>
| |
| | |
| to obtain | |
| | |
| :<math> M = i \left( E - \varepsilon \sin E \right) </math>
| |
| | |
| ===Radial Kepler equation===
| |
| The Radial Kepler equation is:
| |
| {{Equation box 1
| |
| |indent =:
| |
| |equation = <math> t(x) = \sin^{-1}( \sqrt{ x } ) - \sqrt{ x ( 1 - x ) } </math>
| |
| |border colour|background colour}}
| |
| | |
| where ''t'' is time, and ''x'' is the distance along an ''x''-axis.
| |
| This equation is derived by multiplying Kepler's equation by 1/2 making the replacement
| |
| | |
| :<math> E = 2 \sin^{-1}(\sqrt{ x }) </math>
| |
| | |
| and setting ε = 1 gives
| |
| | |
| :<math> t(x) = \frac{1}{2}\left[ E - \sin( E ) \right]. </math>
| |
| | |
| == Inverse problem ==
| |
| | |
| Calculating ''M'' for a given value of ''E'' is straightforward. However, solving for ''E'' when ''M'' is given can be considerably more challenging.
| |
| | |
| Kepler's equation can be solved for ''E'' [[analytic function|analytically]] by [[Lagrange inversion theorem|Lagrange inversion]]. The solution of Kepler's equation given by two Taylor series below.
| |
| | |
| Confusion over the solvability of Kepler's equation has persisted in the literature for four centuries.<ref>It is often erroneously claimed that Kepler's equation "cannot be solved analytically". http://www.jgiesen.de/kepler/kepler.html</ref><ref>Many authors make the absurd claim that it cannot be solved at all. M. V. K. Chari, Sheppard Joel Salon 2000 Technology & Engineering</ref> Kepler himself expressed doubt at the possibility of finding a general solution.
| |
| | |
| {{cquote|I am sufficiently satisfied that it [Kepler's equation] cannot be solved a priori, on account of the different nature of the arc and the sine. But if I am mistaken, and any one shall point out the way to me, he will be in my eyes the great [[Apollonius of Perga|Apollonius]].|20px|20px|Johannes Kepler <ref>Kepler's Problem, by Asaph Hall 1883 Annals of Mathematics</ref>}}
| |
| | |
| ===Inverse Kepler equation===
| |
| The inverse Kepler equation is the solution of Kepler's equation for all real values of ε:
| |
| | |
| : <math>
| |
| E =
| |
| \begin{cases}
| |
| | |
| \displaystyle \sum_{n=1}^\infty
| |
| {\frac{M^{\frac{n}{3}}}{n!}} \lim_{\theta \to 0^+} \! \Bigg(
| |
| \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}} \bigg( \bigg(
| |
| \frac{\theta}{ \sqrt[3]{\theta - \sin(\theta)} } \bigg)^{\!\!\!n} \bigg)
| |
| \Bigg)
| |
| , & \varepsilon = 1 \\
| |
| | |
| \displaystyle \sum_{n=1}^\infty
| |
| { \frac{ M^n }{ n! } }
| |
| \lim_{\theta \to 0^+} \! \Bigg(
| |
| \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}} \bigg( \Big(
| |
| \frac{ \theta }{ \theta - \varepsilon \sin(\theta)} \Big)^{\!n} \bigg)
| |
| \Bigg)
| |
| , & \varepsilon \ne 1
| |
| | |
| \end{cases}
| |
| </math>
| |
| | |
| Evaluating this yields:
| |
| | |
| : <math> | |
| E =
| |
| \begin{cases} \displaystyle
| |
| x + \frac{1}{60} x^3 + \frac{1}{1400}x^5 + \frac{1}{25200}x^7 + \frac{43}{17248000}x^9 + \frac{ 1213}{7207200000 }x^{11} +
| |
| \frac{151439}{12713500800000 }x^{13}+ \cdots \ | \ x = ( 6 M )^\frac{1}{3}
| |
| , & \varepsilon = 1 \\
| |
| \\
| |
| \displaystyle
| |
| \frac{1}{1-\varepsilon} M
| |
| - \frac{\varepsilon}{( 1-\varepsilon)^4 } \frac{M^3}{3!}
| |
| + \frac{(9 \varepsilon^2 + \varepsilon)}{(1-\varepsilon)^7 } \frac{M^5}{5!}
| |
| - \frac{(225 \varepsilon^3 + 54 \varepsilon^2 + \varepsilon ) }{(1-\varepsilon)^{10} } \frac{M^7}{7!}
| |
| + \frac{ (11025\varepsilon^4 + 4131 \varepsilon^3 + 243 \varepsilon^2 + \varepsilon ) }{(1-\varepsilon)^{13} } \frac{M^9}{9!}+ \cdots
| |
| , & \varepsilon \ne 1
| |
| | |
| \end{cases} </math>
| |
| | |
| These series can be reproduced in [[Mathematica]] with the InverseSeries operation.
| |
| : <code>InverseSeries[Series[M - Sin[M], {M, 0, 10}]]</code> | |
| : <code>InverseSeries[Series[M - e Sin[M], {M, 0, 10}]]</code>
| |
| | |
| These functions are simple Taylor series. Taylor series representations of transcendental functions are considered to be definitions of those functions. Therefore this solution is a formal definition of the inverse Kepler equation. While this solution is the simplest in a certain mathematical sense, for values of ε near 1 the convergence is very poor, other solutions are preferable for most applications. Alternatively, Kepler's equation can be solved numerically.
| |
| | |
| The solution for ε ≠ 1 was discovered by [[Karl Stumpff]] in 1968,<ref>Stumpff, Karl (1968b) On The application of Lie-series to the problems of celestial mechanics, NASA Technical Note D-4460</ref> but its significance wasn't recognized.<ref>Colwell, Peter (1993), Solving Kepler's Equation Over Three Centuries, Willmann–Bell. p. 43.</ref>
| |
| | |
| ===Inverse radial Kepler equation===
| |
| The inverse radial Kepler equation is:
| |
| : <math>
| |
| x( t ) = \sum_{n=1}^{ \infty }
| |
| \left[
| |
| \lim_{ r \to 0^+ } \left(
| |
| {\frac{ t^{ \frac{ 2 }{ 3 } n }}{ n! }}
| |
| \frac{\mathrm{d}^{\,n-1}}{\mathrm{ d } r ^{\,n-1}} \! \left(
| |
| r^n \left( \frac{ 3 }{ 2 } \Big( \sin^{-1}( \sqrt{ r } ) - \sqrt{ r - r^2 } \Big)
| |
| \right)^{ \! -\frac{2}{3} n }
| |
| \right) \right)
| |
| \right] </math>
| |
| | |
| Evaluating this yields:
| |
| :<math>x(t) = p - \frac{1}{5} p^2 - \frac{3}{175}p^3 | |
| - \frac{23}{7875}p^4 - \frac{1894}{3931875}p^5 - \frac{3293}{21896875}p^6 - \frac{2418092}{62077640625}p^7 - \ \cdots \
| |
| \bigg| { p = \left( \tfrac{3}{2} t \right)^{2/3} } </math>
| |
| <br>
| |
| To obtain this result using [[Mathematica]]:
| |
| :<code>InverseSeries[Series[ArcSin[Sqrt[t]] - Sqrt[(1 - t) t], {t, 0, 15}]]</code>
| |
| | |
| ==Numerical approximation of inverse problem==
| |
| For most applications, the inverse problem can be computed numerically by finding the [[Zero of a function|root]] of the function:
| |
| | |
| : <math>
| |
| f(E) = E - \varepsilon \sin(E) - M(t)
| |
| </math>
| |
| | |
| This can be done iteratively via [[Newton's method]]:
| |
| | |
| : <math>
| |
| E_{n+1} = E_{n} - \frac{f(E_{n})}{f'(E_{n})} =
| |
| E_{n} - \frac{ E_{n} - \varepsilon \sin(E_{n}) - M(t) }{ 1 - \varepsilon \cos(E_{n})}
| |
| </math>
| |
| | |
| Note that ''E'' and ''M'' are in units of radians in this computation. This iteration is repeated until desired accuracy is obtained (e.g. when ''f''(''E'') < desired accuracy). For most elliptical orbits an initial value of ''E''<sub>0</sub> = ''M''(''t'') is sufficient. For orbits with ''ε'' > 0.8, an initial value of ''E''<sub>0</sub> = ''π'' should be used.<ref>
| |
| {{cite book
| |
| | last1 = Montenbruck
| |
| | first1 = Oliver
| |
| | last2 = Pfleger
| |
| | first2 = Thomas
| |
| | title = Astronomy on the Personal Computer
| |
| | publisher = Springer-Verlag Berlin Heidelberg
| |
| | year = 2009
| |
| | ISBN = 978-3-540-67221-0}}, pp 64–65</ref>
| |
| A similar approach can be used for the hyperbolic form of Kepler's equation. In the case of a parabolic trajectory, Barker's equation is used.
| |
| | |
| == See also ==
| |
| *[[Kepler's laws of planetary motion]]
| |
| *[[Kepler problem]]
| |
| *[[Kepler problem in general relativity]]
| |
| *[[Radial trajectory]]
| |
| | |
| == References ==
| |
| | |
| {{Reflist}}
| |
| | |
| == External links ==
| |
| {{commons category|Kepler's equation}}
| |
| * [http://mathworld.wolfram.com/KeplersEquation.html Kepler's Equation at Wolfram Mathworld]
| |
| | |
| <!--- Categories --->
| |
| | |
| [[Category:Johannes Kepler]]
| |
| [[Category:Orbits]]
| |
Decoration Books is one particular of the finest guides that assist you to decorate your preferred locations and things. In simple words with the assist of decorating book you can decorate your house, garden, office, cakes, cookies, etc. It has the greatest collection in most of the topics that a decorator seeks for. With decorating book you can make a nicely-decorated output. It does not matters whether or not you are a excellent decorator or not but with an suitable decorating book you can decorate simply and quickly.
Decoration Books: Benefits
Decorating book includes skilled decorating ideas, clear directions, lavish photographs and effortless tasks that aid you to decorate with ease. Decoration Books help you to make uniquely decorated house.
There are distinct varieties of decoration books available in the marketplace such as cookie decorating book, interior decorating book and so on. A single of such varieties of painting book is Cake decorating book.
Cake Decorating Book is a ideal decision to decorate the cake. This book offers you important tips, together with illustrated pictures to decorate your birthday cakes, Christmas cake, wedding cake and celebration cake. My uncle learned about book a celebrity by browsing Google Books. If you want to decorate the cake for any special occasion then you require to buy a cake decorating book.
Now days you can locate enormous collection of such books on the web. This book can help you to decorate your cake in a expert manner. Cake Decorating Books not only assist you to decorate your cake but it also aids you to make your cake scrumptious, which is an added benefit. Cake decorating book, which is supplied on the web, requires no basic encounter. With the aid of cake decorating book you can prepare exclusive decorated pastries or cakes. This book is published in such a way that no one can refrain oneself from creating a decorated cake. Every chapter carries new tips and suggestions.
It is essential that one particular must decorate the cake as it gives a distinct and attractive appear that produces the desire to eat it. This cake decorating book is also ideal for novices, which provides step by step instructions and guidance.. Be taught additional resources on our affiliated link - Click here: details. For extra information, consider checking out: like i said.
If you're ready to see more in regards to cobra health insurance review our own web site.