Quaternionic vector space: Difference between revisions
Jump to navigation
Jump to search
en>Giftlite m −sp |
en>David Eppstein |
||
| Line 1: | Line 1: | ||
In [[orbital mechanics]], the '''universal variable formulation''' is a method used to solve the [[two-body problem|two-body]] [[Kepler problem]]. It is a generalized form of [[Kepler's Equation]]s, extending them to apply not only to [[elliptic orbits]], but also [[parabolic orbit|parabolic]] and [[hyperbolic orbit]]s. It thus is applicable to many situations in the [[solar system]], where orbits of widely varying [[orbital eccentricity|eccentricities]] are present. | |||
==Introduction== | |||
A common problem in orbital mechanics is the following: given a body in an [[orbit]] and a time ''t<sub>0</sub>'', find the position of the body at any other given time ''t''. | |||
For [[elliptical orbit]]s with a reasonably small [[Orbital eccentricity|eccentricity]], solving [[Kepler's Equation]] by methods like [[Newton's method]] gives adequate results. However, as the orbit becomes more and more eccentric, the numerical iteration may start to [[limit of a sequence|converge]] slowly or not at all.<ref name=Danby>{{citation |author=Danby, J. M. A.|title=Fundamentals of Celestial Mechanics|publisher=Willman-Bell|year=1988}}</ref> Furthermore, Kepler's equation cannot be applied to [[Parabolic orbit|parabolic]] and [[hyperbolic orbit]]s, since it specifically is tailored to elliptic orbits. | |||
==Derivation== | |||
Although equations similar to Kepler's equation can be derived for parabolic and hyperbolic orbits, it is more convenient to introduce a new independent variable to take the place of the [[eccentric anomaly]] ''E'', and having a single equation that can be solved regardless of the eccentricity of the orbit. The new variable ''s'' is defined by the following [[differential equation]]: | |||
:<math>\frac{ds}{dt} = \frac{1}{r}</math> | |||
where <math>r = r(t)</math> is the time-dependent distance to the center of attraction. The fundamental equation <math>\frac{d^2\mathbf{r}}{dt^2} + \mu \frac{\mathbf{r}}{r^3} = \mathbf{0}</math> is [[regularization|regularized]] by applying this change of variables to yield:<ref name=Danby/> | |||
:<math>\frac{d^2\mathbf{r}}{ds^2} + \alpha\ \mathbf{r} = -\mathbf{P}</math> | |||
where '''P''' is a constant [[Euclidean vector|vector]] and <math>\alpha</math> is defined by | |||
:<math>\alpha = \frac\mu a</math> | |||
The equation is the same as the equation for the [[harmonic oscillator]], a well-known equation in both [[physics]] and [[mathematics]]. Taking the derivative again, we get a third-degree differential equation: | |||
:<math>\frac{d^3\mathbf r} {ds^3} + \alpha\frac{d\mathbf r} {ds} = \mathbf{0}</math> | |||
The family of solutions to this differential equation<ref name="Danby"/> are written symbolically as the functions <math>1,\ s\ c_1(\alpha s^2),\ s^2\ c_2(\alpha s^2),</math> where the functions <math>\ c_k(x)</math>, called [[Stumpff function]]s, are generalizations of sine and cosine functions. Applying this results in:<ref name="Danby">Equation 6.9.26</ref> | |||
:<math>t - t_0 = r_0\ s\ c_1(\alpha s^2) + r_0 \frac{dr_0}{dt}\ s^2\ c_2(\alpha s^2) + \mu \ s^3\ c_3(\alpha s^2)</math> | |||
which is the universal variable formulation of Kepler's Equation. This equation can now be solved numerically using a [[root-finding algorithm]] such as [[Newton's method]] or [[Laguerre's method]] for a given time <math>t</math> to yield <math>s</math>, which in turn is used to compute the [[f and g functions]]: | |||
:<math>\begin{align} | |||
f(s) & = 1 - \left(\frac \mu {r_0}\right) s^2 c_2(\alpha s^2), \\ | |||
g(s) & = t - t_0 - \mu s^3c_3(\alpha s^2), \\ | |||
\frac{df}{dt} & = \dot{f}(s) = -\left(\frac{\mu}{r r_0}\right)s c_1(\alpha s^2), \\ | |||
\frac{dg}{dt} & = \dot{g}(s) = 1 - \left(\frac{\mu}{r}\right)s^2c_2(\alpha s^2) | |||
\end{align}</math> | |||
The values of the f and g functions determine the position of the body at the time <math>t</math>: | |||
:<math>\mathbf{r} = \mathbf{r}_0\ f(s) + \mathbf{v}_0\ g(s)</math> | |||
In addition the velocity of the body at time <math>t</math> can be found using <math>\dot{f}(s)</math> and <math>\dot{g}(s)</math> as follows: | |||
:<math>\mathbf{v} = \mathbf{r}_0\ \dot{f}(s) + \mathbf{v}_0\ \dot{g}(s)</math> | |||
where <math>\mathbf{r}</math> and <math>\mathbf{v}</math> are the position and velocity respectively at time <math>t</math>, and <math>\mathbf{r}_0</math> and <math>\mathbf{v}_0</math> are the position and velocity, respectively, at arbitrary initial time <math>t_0</math>. | |||
==References== | |||
<references/> | |||
[[Category:Orbits]] | |||
Latest revision as of 20:03, 14 February 2013
In orbital mechanics, the universal variable formulation is a method used to solve the two-body Kepler problem. It is a generalized form of Kepler's Equations, extending them to apply not only to elliptic orbits, but also parabolic and hyperbolic orbits. It thus is applicable to many situations in the solar system, where orbits of widely varying eccentricities are present.
Introduction
A common problem in orbital mechanics is the following: given a body in an orbit and a time t0, find the position of the body at any other given time t. For elliptical orbits with a reasonably small eccentricity, solving Kepler's Equation by methods like Newton's method gives adequate results. However, as the orbit becomes more and more eccentric, the numerical iteration may start to converge slowly or not at all.[1] Furthermore, Kepler's equation cannot be applied to parabolic and hyperbolic orbits, since it specifically is tailored to elliptic orbits.
Derivation
Although equations similar to Kepler's equation can be derived for parabolic and hyperbolic orbits, it is more convenient to introduce a new independent variable to take the place of the eccentric anomaly E, and having a single equation that can be solved regardless of the eccentricity of the orbit. The new variable s is defined by the following differential equation:
where is the time-dependent distance to the center of attraction. The fundamental equation is regularized by applying this change of variables to yield:[1]
where P is a constant vector and is defined by
The equation is the same as the equation for the harmonic oscillator, a well-known equation in both physics and mathematics. Taking the derivative again, we get a third-degree differential equation:
The family of solutions to this differential equation[1] are written symbolically as the functions where the functions , called Stumpff functions, are generalizations of sine and cosine functions. Applying this results in:[1]
which is the universal variable formulation of Kepler's Equation. This equation can now be solved numerically using a root-finding algorithm such as Newton's method or Laguerre's method for a given time to yield , which in turn is used to compute the f and g functions:
The values of the f and g functions determine the position of the body at the time :
In addition the velocity of the body at time can be found using and as follows:
where and are the position and velocity respectively at time , and and are the position and velocity, respectively, at arbitrary initial time .
References
- ↑ 1.0 1.1 1.2 1.3 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 Cite error: Invalid<ref>tag; name "Danby" defined multiple times with different content