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| {{Multiple issues
| | As the economic situation becomes more and more stressful, you need to have other options for investments, this is where bullion coins come in. There are lots of options available, when you want to purchase bullion coins. All you need to do is some research and have the willingness to make that jump into these investments.<br><br>When decorating for the party, remember to think green! Look for green decorations such as balloons, streamers and plastic tablecloths. You can also purchase fun cardboard cut-outs that say "Happy St. Patrick's Day." For an Irish flag theme, you can choose to display a few orange, green and white decorations.<br><br><br><br>This tossing game is plenty of fun. Simply set up different size black pots and have the children take turns tossing gold coins into the pot. Every time the coins end up in a pot, the child earns points. Point value should depend on the size and distance of the pot.<br><br>With the many choices that can be found for silver coins, you are likely to want to purchase from someone who offers you the highest quality. However when you are buying silver coins you also want good customer service as well. After all, you are going to be spending a large amount of money on your silver coins and therefore you should be [http://mondediplo.com/spip.php?page=recherche&recherche=treated treated] with great respect.<br><br>Besides purity, another important characteristic is its weight. Normally, there are 4 sizes for gold bullion coins, but the 1 oz bullion coin is the most frequently used.<br><br>When you buy silver bullion, it can be a start of a growing personal investment. Start with coins which are made of silver. They are usually cheaper compared to gold coins, but their value increases as the time goes by. The value of silver coins depends on its rarity. If you have a silver coin which dates back more than 80 years ago, then you've hit a jackpot. There weren't too many silver coins that were produced during those times. It's crucial to note that once melted, silver coins can value up to 20 or even 50 times their face value. Once assessed by appraisers and numismatists they can even buy them from you. Make a research on old coins which you in possession. This will make you aware of the [http://coins.goldgrey.org/gold-bullions/ http://coins.goldgrey.org/gold-bullions/] current price of rare silver coins, so you won't get fooled once you decide to sell them.<br><br>Does this mean that you should rush out and purchase gold jewelry? Should you "invest in gold" through traditional means? Actually, the answer is both yes and no. Few people today remember that not long ago, it was illegal to own gold bullion. Fewer still recall that Roosevelt actually confiscated the gold owned by the American people following the Great Depression. Does this mean that investing in gold is not a good idea?<br><br>Before you invest in silver, you have to make an effort to study the market and the business itself. What you have to keep in mind is that silver is different from paper assets. In this case, you have to perform due diligence at research. There are speculations surrounding this type of investment. Included in such speculations is the rising cost of this precious metal. The right education can help you move forward and get ahead in this investment. You have to make sure that you know just how the silver investment works. In this case, you will need the assistance of a professional who can teach you what is involved in the entire process of buying and selling. This way, you can know just what you need to do.<br><br>What makes this particular coin so important? For many, it is a collector's item, an important piece to have in your collection if you are a [http://Www.twitpic.com/tag/coin+collector coin collector]. However, it is far more than that. With the value of gold on the rise, the gold American Eagle is a powerful investment tool. Many who now buy it do so because of its significance in terms of value. This makes it one of the most sought after of all coins, because a trusted authority backs it and it's considered one of the highest quality bullion coins available today. As you consider an investment into this precious metal, doing so with the use of this coin may be the ideal route to take. |
| | technical = December 2011
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| | one source = December 2011
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| | more footnotes = December 2011
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| }}
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| '''Orbital perturbation analysis''' is the activity of determining why a [[satellite|satellite's]] orbit differs from the mathematical ideal orbit. A satellite's [[Kepler orbit|orbit]] in an ideal two-body system describes a conic section, or ellipse. In reality, there are several factors that cause the conic section to continually change. These deviations from the ideal Kepler's orbit are called [[Perturbation (astronomy)|perturbations]].
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| ==Perturbation of spacecraft orbits==
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| It has long been recognized that the [[Lunar theory|Moon]] does not follow a perfect orbit, and many theories and models have been examined over the millennia to explain it. [[Isaac Newton]] determined the primary contributing factor to orbital perturbation of the moon was that the shape of the Earth is actually an [[oblate spheroid]] due to its spin, and he used the perturbations of the lunar orbit to estimate the oblateness of the Earth.
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| In Newton's [[Philosophiæ Naturalis Principia Mathematica]], he demonstrated that the gravitational force between two mass points is inversely proportional to the square of the distance between the points, and he fully solved the corresponding [[Kepler orbit|"two-body problem"]] demonstrating that the radius vector between the two points would describe an ellipse. But no exact closed analytical form could be found for the [[three body problem]]. Instead, mathematical models called "orbital perturbation analysis" have been developed. With these techniques a quite accurate mathematical description of the trajectories of all the planets could be obtained. Newton recognized that the Moon's perturbations could not entirely be accounted for using just the solution to the three body problem, as the deviations from a pure Kepler orbit around the Earth are much larger than deviations of the orbits of the planets from their own Sun-centered Kepler orbits, caused by the gravitational attraction between the planets. With the availability of digital computers and the ease with which we can now compute orbits, this problem has partly disappeared, as the motion of all celestial bodies including planets, satellites, asteroids and comets can be modeled and predicted with almost perfect accuracy using the method of the numerical propagation of the trajectories. Nevertheless several analytical closed form expressions for the effect of such additional "perturbing forces" are still very useful.
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| All celestial bodies of the [[Solar System]] follow in first approximation a [[Kepler orbit]] around a central body. For a satellite (artificial or natural) this central body is a planet. But both due to gravitational forces caused by the Sun and other celestial bodies and due to the flattening of its planet (caused by its rotation which makes the planet slightly oblate and therefore the result of the [[Shell theorem]] not fully applicable) the satellite will follow an orbit around the Earth that deviates more than the Kepler orbits observed for the planets.
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| The precise modeling of the motion of the Moon has been a difficult task. The best and most accurate modeling for the lunar orbit before the availability of digital computers was obtained with the complicated [[Charles-Eugène Delaunay|Delaunay]] and [[Ernest William Brown|Brown]]'s [[lunar theory|lunar theories]].
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| For man-made spacecraft orbiting the Earth at comparatively low altitudes the deviations from a Kepler orbit are much larger than for the Moon. The approximation of the gravitational force of the Earth to be that of a homogeneous sphere gets worse the closer one gets to the Earth surface and the majority of the artificial Earth satellites are in orbits that are only a few hundred kilometers over the Earth surface. Furthermore they are (as opposed to the Moon) significantly affected by the solar radiation pressure because of their large cross-section to mass ratio; this applies in particular to [[3-axis stabilized spacecraft]] with large [[solar arrays]]. In addition they are significantly affected by rarefied air below 800–1000 km. The air drag at high altitudes is also dependent on [[Space weather|solar activity]]. | |
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| ==Mathematical approach==
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| Consider any function
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| :<math>g(x_1,x_2,x_3,v_1,v_2,v_3)\,</math>
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| of the position
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| :<math>x_1,x_2,x_3\,</math>
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| and the velocity | |
| :<math>v_1,v_2,v_3\,</math>
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| From the chain rule of differentiation one gets that the time derivative of <math>g</math> is
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| :<math>\dot{g}\ =\ \frac{\partial g }{\partial x_1}\ v_1\ + \ \frac{\partial g }{\partial x_2}\ v_2\ + \frac{\partial g }{\partial x_3}\ v_3\ + \ \frac{\partial g }{\partial v_1}\ f_1\ + \ \frac{\partial g }{\partial v_2}\ f_2\ + \ \frac{\partial g }{\partial v_3}\ f_3</math>
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| where <math>f_1\ ,\ f_2\ ,\ f_3</math> are the components of the force per unit mass acting on the body.
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| If now <math>g</math> is a "constant of motion" for a [[Kepler orbit]] like for example an [[orbital element]] and the force is corresponding "Kepler force"
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| :<math>
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| (f_1\ ,\ f_2\ ,\ f_3)\ = \ - \frac {\mu} {r^3}\ (x_1\ ,\ x_2\ ,\ x_3)
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| </math>
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| one has that <math>\dot{g}\ =\ 0\,</math>.
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| If the force is the sum of the "Kepler force" and an additional force (force per unit mass)
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| :<math>(h_1\ ,\ h_2\ ,\ h_3)</math>
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| i.e.
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| :<math>
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| (f_1\ ,\ f_2\ ,\ f_3)\ = \ - \frac {\mu} {r^3}\ (x_1\ ,\ x_2\ ,\ x_3)\ +\ (h_1\ ,\ h_2\ ,\ h_3)
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| </math> | |
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| one therefore has
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| :<math>\dot{g}\ =\frac{\partial g }{\partial v_1}\ h_1\ + \ \frac{\partial g }{\partial v_2}\ h_2\ + \ \frac{\partial g }{\partial v_3}\ h_3</math>
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| and that the change of <math>g\,</math> in the time from <math>t=t_1\,</math> to <math>t=t_2\,</math> is
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| :<math>
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| \Delta g\ =\ \int\limits_{t_1}^{t_2}\left(\frac{\partial g }{\partial v_1}\ h_1\ + \ \frac{\partial g }{\partial v_2}\ h_2\ + \ \frac{\partial g }{\partial v_3}\ h_3 \right)dt
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| </math>
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| If now the additional force <math>(h_1\ ,\ h_2\ ,\ h_3)\,</math> is sufficiently small that the motion will be close to that of a [[Kepler orbit]] one gets an approximate value for <math>\Delta g\,</math> by evaluating this integral assuming
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| <math>x_1(t),x_2(t),x_3(t)\,</math> to precisely follow this [[Kepler orbit]].
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| In general one wants to find an approximate expression for the change <math>\Delta g\,</math> over one orbital revolution using the true anomaly <math>\theta\,</math> as integration variable, i.e. as
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| # {{NumBlk|:|<math>
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| \Delta g\ =\ \int\limits_{0}^{2\pi}\left(\frac{\partial g }{\partial v_1}\ h_1\ + \ \frac{\partial g }{\partial v_2}\ h_2\ + \ \frac{\partial g }{\partial v_3}\ h_3 \right)\frac{r^2}{\sqrt{\mu p}}d\theta
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| </math>|{{EquationRef|1}}}}
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| This integral is evaluated setting <math>r(\theta)=\frac {p}{1+e\cos \theta}\,</math>, the elliptical Kepler orbit in polar angles.
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| For the transformation of integration variable from time to [[true anomaly]] it was used that the angular momentum <math>H\ =\ r^2\ \dot{\theta}\ =\ \sqrt{\mu p} \,</math> by definition of the parameter <math>p\,</math> for a Kepler orbit (see equation (13) of the [[Kepler orbit]] article).
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| For the special case where the Kepler orbit is circular or almost circular
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| :<math>r\ =\ p</math> and ({{EquationNote|1}}) takes the simpler form | |
| # {{NumBlk|:|<math>
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| \Delta g\ =\ \frac{P}{2\pi}\ \int\limits_{0}^{2\pi}\left(\frac{\partial g }{\partial v_1}\ h_1\ + \ \frac{\partial g }{\partial v_2}\ h_2\ + \ \frac{\partial g }{\partial v_3}\ h_3 \right)d\theta
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| </math>|{{EquationRef|2}}}}
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| where <math>P\ =\ 2\pi\ r\ \sqrt{\frac{r}{\mu}}\,</math> is the orbital period
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| ===Perturbation of the semi-major axis/orbital period===
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| For an elliptic [[Kepler orbit]], the sum of the kinetic and the potential energy
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| :<math>g = \frac{V^2}{2}-\frac {\mu} {r}</math>,
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| where <math>V\,</math> is the orbital velocity, is a constant and equal to
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| :<math>g\ =\ -\frac {\mu} {2 \cdot a}</math> (Equation ([[Kepler_orbit#equation_44|44]]) of the Kepler orbit article)
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| If <math>\bar{h}\,</math> is the perturbing force and <math>\bar{V}\,</math>is the velocity vector of the Kepler orbit the equation ({{EquationNote|1}}) takes the form:
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| # {{NumBlk|:|<math>
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| \Delta g\ =\ \int\limits_{0}^{2\pi}\bar{V} \bar{h}\frac{r^2}{\sqrt{\mu p}}d\theta
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| </math>|{{EquationRef|3}}}} | |
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| and for a circular or almost circular orbit
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| # {{NumBlk|:|<math>
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| \Delta g\ =\ \frac{P}{2\pi}\ \int\limits_{0}^{2\pi} \bar{V} \bar{h}d\theta
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| </math>|{{EquationRef|4}}}}
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| From the change <math>\Delta g\,</math> of the parameter <math>g\,</math> the new semi-major axis <math>a\,</math> and the new period <math>P\ =\ 2\pi\ a\ \sqrt{\frac{a}{\mu}}\,</math> are computed (relations (43) and (44) of the [[Kepler orbit]] article).
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| ===Perturbation of the orbital plane===
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| Let <math>\hat{g}\,</math> and <math>\hat{h}\,</math> make up a rectangular coordinate system in the plane of the reference Kepler orbit. If <math>\omega\,</math> is the argument of perigee relative the <math>\hat{g}\,</math> and <math>\hat{h}\,</math> coordinate system the true anomaly <math>\theta\,</math> is given by <math>\theta=u-\omega\,</math> and the approximate change <math> \Delta \hat{z}\,</math> of the orbital pole <math> \hat{z}\,</math> (defined as the unit vector in the direction of the angular momentum) is
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| # {{NumBlk|:|<math>
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| \Delta \hat{z}\ =\ \int\limits_{0}^{2\pi}\frac{f_z }{V_t} (\hat{g} \cos u + \hat{h} \sin u)\frac{r^2}{\sqrt{\mu p}}du \quad \times \ \hat{z}
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| =\ \frac{1}{\mu p}\left[\hat{g}\int\limits_{0}^{2\pi}f_z r^3 \cos u \ du
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| +\ \hat{h}\int\limits_{0}^{2\pi}f_z r^3 \sin u \ du \right]\quad \times \ \hat{z}
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| </math>|{{EquationRef|5}}}}
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| where <math>f_z\,</math> is the component of the perturbing force in the <math> \hat{z}\,</math> direction, <math>V_t=\sqrt{\frac{\mu}{p}}\ (1+e\ \cos\theta)\,</math> is the velocity component of the Kepler orbit orthogonal to radius vector and <math>r=\frac{p}{1+e\ \cos\theta}\,</math> is the distance to the center of the Earth.
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|
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| For a circular or almost circular orbit ({{EquationNote|5}}) simplifies to
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| # {{NumBlk|:|<math>
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| \Delta \hat{z}\ =\ \frac{r^2}{\mu}\left[\hat{g}\int\limits_{0}^{2\pi}f_z \cos u \ du
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| +\ \hat{h}\int\limits_{0}^{2\pi}f_z \sin u \ du \right]\quad \times \ \hat{z}
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| </math>|{{EquationRef|6}}}}
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| '''Example'''
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| In a circular orbit a low-force propulsion system ([[Ion thruster]]) generates a thrust (force per unit mass) of <math>F\ \hat{z}\,</math> in the direction of the orbital pole in the half of the orbit for which <math>\sin u\,</math> is positive and in the opposite direction in the other half. The resulting change of orbit pole after one orbital revolution of duration <math>P\ =\ 2\pi\ r\ \sqrt{\frac{r}{\mu}}\,</math> is
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| # {{NumBlk|:|<math>
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| \Delta \hat{z}\ =\ \frac{r^2}{\mu}\left[\ 2\ F\int\limits_{0}^{\pi}\sin u \ du \right]\quad \hat{h}\times \hat{z} =
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| \ \frac{r^2}{\mu}\ 4\ F\ \quad \hat{g}
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| </math>|{{EquationRef|7}}}}
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| The average change rate <math>\frac{\Delta \hat{z}}{P}\,</math> is therefore
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| # {{NumBlk|:|<math>
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| \frac{\Delta \hat{z}}{P} =\ \frac{2}{\pi}\ \frac{F}{V}\ \hat{g}
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| </math>|{{EquationRef|8}}}}
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| where <math>V\ =\ \sqrt{\frac{\mu}{r}},</math> is the orbital velocity in the circular Kepler orbit.
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| ===Perturbation of the eccentricity vector===
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| Rather than applying (1) and (2) on the partial derivatives of the orbital elements '''eccentricity''' and '''argument of perigee''' directly one should apply these relations for the [[eccentricity vector]]. First of all the typical application is a near-circular orbit. But there are also mathematical advantages working with the partial derivatives of the components of this vector also for orbits with a significant eccentricity.
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| Equations (60), (55) and (52) of the [[Kepler orbit]] article say that the eccentricity vector is
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|
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| # {{NumBlk|:|<math>\bar{e}=\frac{(V_t-V_0) \cdot \hat{r} - V_r \cdot \hat{t}}{V_0}</math>|{{EquationRef|9}}}}
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| where
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| # {{NumBlk|:|<math>V_0 = \sqrt{\frac{\mu}{p}}</math>|{{EquationRef|10}}}}
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| # {{NumBlk|:|<math>p = \frac{{(r \cdot V_t)}^2}{\mu }</math>|{{EquationRef|11}}}}
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| from which follows that
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| # {{NumBlk|:|<math>\frac{\partial\bar{e}}{\partial V_r} = -\frac {1}{V_0} \hat{t}</math>|{{EquationRef|12}}}}
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| # {{NumBlk|:|<math>\frac{\partial\bar{e}}{\partial V_t} = \frac {1}{V_0} \left(2\ \hat{r}-\frac{V_r}{V_t}\ \hat{t}\right)</math>|{{EquationRef|13}}}}
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| where
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| # {{NumBlk|:|<math>V_r = \sqrt{\frac {\mu}{p}} \cdot e \cdot \sin \theta</math>|{{EquationRef|14}}}}
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| # {{NumBlk|:|<math>V_t = \sqrt{\frac {\mu}{p}} \cdot (1 + e \cdot \cos \theta)</math>|{{EquationRef|15}}}}
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| (Equations (18) and (19) of the [[Kepler orbit]] article)
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| The eccentricity vector is by definition always in the [[Osculating orbit|osculating]] orbital plane spanned by <math>\hat{r}</math> and <math>\hat{t}</math> and formally there is also a derivative
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| :<math>\frac{\partial\bar{e}}{\partial V_z} = -\frac {V_r}{V_0}\ \frac{\partial\hat{t}}{\partial V_z}</math>
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| with
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| :<math>\frac{\partial\hat{t}}{\partial V_z} = \frac {1}{V_t}\ \hat{z}</math>
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| corresponding to the rotation of the orbital plane
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| But in practice the in-plane change of the eccentricity vector is computed as
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| # {{NumBlk|:|<math>
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| \begin{align}
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| \Delta \bar{e}\ = &\frac {1}{V_0}\ \int\limits_{0}^{2\pi}\left(-\hat{t}\ f_r\ + \ \left(2\ \hat{r}-\frac{V_r}{V_t}\ \hat{t}\right)\ f_t\right)\frac{r^2}{\sqrt{\mu p}}du\ = \\
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| &\frac {1}{\mu}\ \int\limits_{0}^{2\pi}\left(-\hat{t}\ f_r\ + \ \left(2\ \hat{r}-\frac{V_r}{V_t}\ \hat{t}\right)\ f_t\right) r^2 du \end{align}
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| </math>|{{EquationRef|16}}}} | |
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| ignoring the out-of-plane force and the new eccentricity vector
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| :<math>\bar{e} + \Delta \bar{e}</math>
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| is subsequently projected to the new orbital plane orthogonal to the new orbit normal | |
| :<math>\hat{z} + \Delta \hat{z}</math>
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| computed as described above.
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| '''Example'''
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| The Sun is in the orbital plane of a spacecraft in a circular orbit with radius <math>r\,</math> and consequently with a constant orbital velocity <math>V_0\ =\ \sqrt{\frac{\mu}{r}}</math> . If <math>\hat{k}\,</math> and <math>\hat{l}\,</math> make up a rectangular coordinate system in the orbital plane such that <math>\hat{k}\,</math> points to the Sun and assuming that the solar radiation pressure force per unit mass <math>F\,</math> is constant one gets that
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| :<math>\hat{r}=\cos(u)\ \hat{k}\ +\ \sin(u)\ \hat{l}\,</math>
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| :<math>\hat{t}=-\sin(u)\ \hat{k}\ +\ \cos(u)\ \hat{l}\,</math>
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| :<math>F_r=-\cos(u)\ F\,</math>
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| :<math>F_t= \sin(u)\ F\,</math>
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| where <math>u\,</math> is the polar angle of <math>\hat{r}\,</math> in the <math>\hat{k}\,</math>, <math>\hat{l}\,</math> system. Applying ({{EquationNote|2}}) one gets that
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| # {{NumBlk|:|<math>
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| \begin{align}
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| \Delta \hat{e}\ & =\ \frac{P}{2\pi}\ \frac {1}{V_0}\ \int\limits_{0}^{2\pi}\left( (-\sin(u)\ \hat{k}\ +\ \cos(u)\ \hat{l}) \ F\ \cos(u)\ + \ 2\ (\cos(u)\ \hat{k}\ +\ \sin(u)\ \hat{l})\ F\ \sin(u)\right)\ du \\
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| & = P\ \frac{3}{2}\ \frac {1}{V_0}\ \ F\ \hat{l}
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| \end{align}
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| </math>|{{EquationRef|17}}}} | |
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| This means the eccentricity vector will gradually increase in the direction <math>\hat{l}\,</math> orthogonal to the Sun direction. This is true for any orbit with a small eccentricity, the direction of the small eccentricity vector does not matter. As <math>P\,</math> is the orbital period this means that the average rate of this increase will be
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| <math>\frac{3}{2}\ \frac {F}{V_0}\,</math>
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| ==The effect of the Earth flattening==
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| [[File:Spherical coordinates unit vectors.svg|thumb|right|Figure 1: The unit vectors <math>\hat{\phi}\ ,\ \hat{\lambda}\ ,\ \hat{r}</math>]]
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| In the article [[Geopotential model]] the modeling of the gravitational field as a sum of spherical harmonics is discussed. By far, the dominating term is the "J2-term". This is a "zonal term" and corresponding force is therefore completely in a longitudinal plane with one component <math>F_r\ \hat{r}\,</math> in the radial direction and one component <math>F_\lambda\ \hat{\lambda}\,</math> with the unit vector <math>\hat{\lambda}\,</math> orthogonal to the radial direction towards north. These directions <math>\hat{r}\,</math> and <math>\hat{\lambda}\,</math> are illustrated in Figure 1. | |
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| [[File:Zonal term force components.svg|thumb|right|Figure 2: The unit vector <math>\hat{t}\,</math> orthogonal to <math>\hat{r}\,</math> in the direction of motion and the orbital pole <math>\hat{z}\,</math>. The force component <math>F_\lambda</math> is marked as "F"]]
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| To be able to apply relations derived in the previous section the force component <math>F_\lambda\ \hat{\lambda}\,</math> must be split into two orthogonal components <math>F_t\ \hat{t}</math> and <math>F_z\ \hat{z}</math> as illustrated in figure 2
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| Let <math>\hat{a}\ ,\ \hat{b}\ ,\ \hat{n}\,</math> make up a rectangular coordinate system with origin in the center of the Earth (in the center of the [[Reference ellipsoid]]) such that <math>\hat{n}\,</math> points in the direction north and such that <math>\hat{a}\ ,\ \hat{b}\,</math> are in the equatorial plane of the Earth with <math>\hat{a}\,</math> pointing towards the [[Orbital node|ascending node]], i.e. towards the blue point of Figure 2.
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| The components of the unit vectors
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| :<math>\hat{r}\ ,\ \hat{t}\ ,\ \hat{z}\,</math>
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| making up the local coordinate system (of which <math>\hat{t}\ ,\ \hat{z},</math> are illustrated in figure 2) relative the <math>\hat{a}\ ,\ \hat{b}\ ,\ \hat{n}\,</math> are
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| :<math>r_a= \cos u\,</math>
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| :<math>r_b= \cos i \ \sin u\,</math>
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| :<math>r_n= \sin i \ \sin u\,</math>
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| :<math>t_a=-\sin u\,</math>
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| :<math>t_b= \cos i \ \cos u\,</math>
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| :<math>t_n= \sin i \ \cos u\,</math>
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| :<math>z_a= 0\,</math>
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| :<math>z_b=-\sin i\,</math>
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| :<math>z_n= \cos i\,</math>
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| where <math>u\,</math> is the polar argument of <math>\hat{r}\,</math> relative the orthogonal unit vectors <math>\hat{g}=\hat{a}\,</math> and <math>\hat{h}=\cos i\ \hat{b}\ +\ \sin i\ \hat{n}\,</math> in the orbital plane
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| Firstly
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| :<math>\sin \lambda =\ r_n\ =\ \sin i \ \sin u\,</math>
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| where <math>\lambda\,</math> is the angle between the equator plane and <math>\hat{r}\,</math> (between the green points of figure 2) and from equation (12) of the article [[Geopotential model]] one therefore gets that
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| # {{NumBlk|:|<math>
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| f_r = J_2\ \frac{1}{r^4}\ \frac{3}{2}\ \left(3\ \sin^2 i\ \sin^2 u\ -\ 1\right)
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| </math>|{{EquationRef|18}}}}
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| Secondly the projection of direction north, <math>\hat{n}\,</math>, on the plane spanned by <math>\hat{t}\ ,\ \hat{z},</math> is
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| | |
| :<math>\sin i \ \cos u \ \hat{t}\ +\ \cos i \ \hat{z}\,</math>
| |
| | |
| and this projection is
| |
| :<math>\cos \lambda \ \hat{\lambda}\,</math>
| |
| | |
| where <math>\hat{\lambda}\,</math> is the unit vector <math>\hat{\lambda}</math> orthogonal to the radial direction towards north illustrated in figure 1.
| |
| | |
| From equation (12) of the article [[Geopotential model]] one therefore gets that
| |
| | |
| :<math>f_\lambda \ \hat{\lambda}\ =\ -J_2\ \frac{1}{r^4}\ 3\ \sin\lambda\ (\sin i \ \cos u \ \hat{t}\ +\ \cos i \ \hat{z}) =\ -J_2\ \frac{1}{r^4}\ 3\ \sin i \ \sin u\ (\sin i \ \cos u \ \hat{t}\ +\ \cos i \ \hat{z})\,</math> | |
| | |
| and therefore:
| |
| # {{NumBlk|:|<math>
| |
| f_t =\ -J_2\ \frac{1}{r^4}\ 3\ \sin^2 i\ \sin u\ \cos u
| |
| </math>|{{EquationRef|19}}}}
| |
| # {{NumBlk|:|<math>
| |
| f_z =\ -J_2\ \frac{1}{r^4}\ 3\ \sin i\ \cos i\ \sin u
| |
| </math>|{{EquationRef|20}}}}
| |
| | |
| ===Perturbation of the orbital plane===
| |
| From ({{EquationNote|5}}) and ({{EquationNote|20}}) one gets that
| |
| | |
| # {{NumBlk|:|<math>
| |
| \Delta \hat{z}\ =\ -J_2\ \frac{3\ \sin i\ \cos i}{\mu p^2}\left[\hat{g}\int\limits_{0}^{2\pi}\frac{p}{r}\ \sin u\ \cos u \ du
| |
| +\ \hat{h}\int\limits_{0}^{2\pi}\frac{p}{r}\ \sin^2 u\ du \right]\quad \times \ \hat{z} | |
| </math>|{{EquationRef|21}}}}
| |
| | |
| The fraction <math>\frac{p}{r}\,</math> is
| |
| :<math>\frac{p}{r}\ =\ 1\ +\ e\ \cos (u-\omega)\ =\ 1\ +\ e\ \cos u\ \cos\omega\ +\ e\ \sin u\ \sin\omega\,</math>
| |
| where <math>e\,</math> is the eccentricity
| |
| and <math>\omega\,</math> is the argument of perigee
| |
| of the reference [[Kepler orbit]] | |
| | |
| As all integrals of type
| |
| :<math>\int\limits_{0}^{2\pi} \cos^m u \ \sin^n u\ du\,</math>
| |
| are zero if not both <math>n\,</math> and <math>m\,</math> are even one gets from ({{EquationNote|21}}) that
| |
| | |
| :<math>
| |
| \Delta \hat{z}\ =\ -2\pi\ \frac{J_2}{\mu\ p^2}\ \frac{3}{2}\ \sin i\ \cos i\ \quad \hat{h} \times \hat{z}
| |
| </math>
| |
| | |
| As
| |
| :<math>
| |
| \hat{n}\ =\ \cos i\ \hat{z}\ + \sin i\ \hat{h}
| |
| </math>
| |
| | |
| this can be written
| |
| # {{NumBlk|:|<math>
| |
| \Delta \hat{z}\ =\ -2\pi\ \frac{J_2}{\mu\ p^2}\ \frac{3}{2}\ \cos i\ \quad \hat{n} \times \hat{z}
| |
| </math>|{{EquationRef|22}}}}
| |
| | |
| As <math>\hat{n}</math> is an inertially fixed vector (the direction of the spin axis of the Earth) relation ({{EquationNote|22}}) is the equation of motion for a unit vector <math>\hat{z}\,</math> describing a cone around <math>\hat{n}</math> with a precession rate (radians per orbit) of <math>-2\pi\ \frac{J_2}{\mu\ p^2}\ \frac{3}{2}\ \cos i\,</math>
| |
| | |
| In terms of orbital elements this is expressed as
| |
| | |
| # {{NumBlk|:|<math>
| |
| \Delta i\ =\ 0
| |
| </math>|{{EquationRef|23}}}}
| |
| # {{NumBlk|:|<math>
| |
| \Delta \Omega\ =\ -2\pi\ \frac{J_2}{\mu\ p^2}\ \frac{3}{2}\ \cos i
| |
| </math>|{{EquationRef|24}}}}
| |
| | |
| where
| |
| :<math>i\,</math> is the inclination of the orbit to the equatorial plane of the Earth
| |
| | |
| :<math>\Omega\,</math> is the right ascension of the ascending node
| |
| | |
| ===Perturbation of the eccentricity vector===
| |
| | |
| From ({{EquationNote|16}}), ({{EquationNote|18}}) and ({{EquationNote|19}}) follows that in-plane perturbation of the eccentricity vector is
| |
| | |
| # {{NumBlk|:|<math>
| |
| \Delta \bar{e}\ =\ \frac {J_2}{\mu\ p^2}\ \int\limits_{0}^{2\pi}\left(-\hat{t}\ \left(\frac{p}{r}\right)^2\ \frac{3}{2}\ \left(3\ \sin^2 i\ \sin^2 u\ -\ 1\right)\ - \ \left(2\ \hat{r}-\frac{V_r}{V_t}\ \hat{t}\right)\ \left(\frac{p}{r}\right)^2\ 3\ \sin^2 i \cos u\ \sin u\right)du
| |
| </math>|{{EquationRef|25}}}}
| |
| | |
| the new eccentricity vector being the projection of
| |
| :<math>\bar{e}+\Delta \bar{e}</math>
| |
| | |
| on the new orbital plane orthogonal to
| |
| :<math>\hat{z}+\Delta \hat{z}</math>
| |
| where <math>\Delta \hat{z}\,</math> is given by ({{EquationNote|22}})
| |
| | |
| Relative the coordinate system
| |
| :<math>\hat{g}=\hat{a}\,</math>
| |
| :<math>\hat{h}=\cos i\ \hat{b}\ +\ \sin i\ \hat{n}\,</math>
| |
| | |
| one has that | |
| | |
| :<math>\hat{r}=\cos u\ \hat{g}\ +\ \sin u\ \hat{h}\,</math>
| |
| :<math>\hat{t}=-\sin u\ \hat{g}\ +\ \cos u\ \hat{h}\,</math>
| |
| | |
| Using that
| |
| | |
| :<math>\frac {p}{r}\ =\ 1 + e \cdot \cos \theta\ =\ 1 + e_g \cdot \cos u + e_h \cdot \sin u</math>
| |
| and that
| |
| | |
| :<math>\frac {V_r}{V_t} = \frac {e_g \cdot \sin u\ -\ e_h \cdot \cos u}{\frac {p}{r}}</math>
| |
| | |
| where
| |
| | |
| :<math>e_g =\ e\ \cos \omega</math>
| |
| :<math>e_h =\ e\ \sin \omega</math>
| |
| | |
| are the components of the eccentricity vector in the <math>\hat{g}\ ,\ \hat{h}\,</math> coordinate system this integral ({{EquationNote|25}}) can be evaluated analytically, the result is
| |
| | |
| # {{NumBlk|:|<math>
| |
| \Delta \bar{e}\ =\ -2\pi\ \frac {J_2}{\mu\ p^2}\ \frac{3}{2} \left(\frac{3}{2}\ \sin^2 i\ -\ 1\right)\ \left(-e_h \hat{g}\ +\ e_g \hat{h}\right)\ =\ -2\pi\ \frac {J_2}{\mu\ p^2} \frac{3}{2} \left(\frac{3}{2}\ \sin^2 i\ -\ 1\right)\ \hat{z}\ \times \ \bar{e}
| |
| </math>|{{EquationRef|26}}}}
| |
| | |
| This the difference equation of motion for the eccentricity vector <math>\bar{e}\,</math> to form a circle, the magnitude of the eccentricity <math>e\,</math> staying constant.
| |
| | |
| Translating this to orbital elements it must be remembered that the new eccentricity vector obtained by adding <math>\Delta \bar{e}\ \,</math> to the old <math>\bar{e}\ \,</math> must be projected to the new orbital plane obtained by applying ({{EquationNote|23}}) and ({{EquationNote|24}})
| |
| | |
| [[File:J2-perturbation.svg|thumb|right|Figure 3: The change <math>\Delta\omega\,</math> in "argument of perigee" after one orbit is the sum of a contribution <math>\Delta \omega_1\,</math> caused by the in-plane force components and a contribution <math>\Delta \omega_2\,</math> caused by the use of the ascending node as reference]]
| |
| This is illustrated in figure 3:
| |
| | |
| To the change in argument of the eccentricity vector
| |
| | |
| :<math>\Delta \omega_1\ =\ -2\pi\ \frac {J_2}{\mu\ p^2}\ \frac{3}{2} \left(\frac{3}{2}\ \sin^2 i\ -\ 1\right)\,</math>
| |
| | |
| must be added an increment due to the precession of the orbital plane (caused by the out-of-plane force component) amounting to
| |
| | |
| :<math>\Delta \omega_2\ =\ -\cos i\ \Delta\Omega \ =\ 2\pi\ \frac {J_2}{\mu\ p^2}\ \frac{3}{2}\ \cos^2 i\,</math>
| |
| | |
| One therefore gets that
| |
| | |
| # {{NumBlk|:|<math>
| |
| \Delta e\ =0
| |
| </math>|{{EquationRef|27}}}}
| |
| | |
| # {{NumBlk|:|<math>
| |
| \Delta \omega\ =\Delta \omega_1\ +\ \Delta \omega_2\ =\ \ -2\pi\ \frac {J_2}{\mu\ p^2}\ 3 \left(\frac{5}{4}\ \sin^2 i\ -\ 1\right)</math>|{{EquationRef|28}}}}
| |
| | |
| In terms of the components of the eccentricity vector <math>e_g,e_h\,</math> relative the coordinate system <math>\hat{g} ,\hat{h}\,</math> that precesses around the polar axis of the Earth the same is expressed as follows
| |
| | |
| # {{NumBlk|:|<math>
| |
| \begin{align}
| |
| &(\Delta e_g,\Delta e_h)\ = \\
| |
| &-2\pi\ \frac {J_2}{\mu\ p^2}\ \frac{3}{2} \left(\frac{3}{2}\ \sin^2 i\ -\ 1\right)\ (-e_h ,e_g)\ + \ 2\pi\ \frac {J_2}{\mu\ p^2}\ \frac{3}{2}\ \cos^2 i\ (-e_h ,e_g ) = \\
| |
| &-2\pi\ \frac {J_2}{\mu\ p^2}\ 3 \left(\frac{5}{4}\ \sin^2 i\ -\ 1\right)\ (-e_h ,e_g)
| |
| \end{align}
| |
| </math>|{{EquationRef|29}}}}
| |
| | |
| where the first term is the in-plane perturbation of the eccentricity vector and the second is the effect of the new position of the ascending node in the new plane
| |
| | |
| From ({{EquationNote|28}}) follows that <math>\Delta \omega\,</math> is zero if <math>\sin^2 i\ =\frac{4}{5}\,</math>. This fact is used for [[Molniya orbit]]s having an inclination of 63.4 deg. An orbit with an inclination of 180 - 63.4 deg = 116.6 deg would in the same way have a constant argument of perigee.
| |
| | |
| ====Proof====
| |
| | |
| Proof that the integral
| |
| | |
| # {{NumBlk|:|<math>
| |
| \int\limits_{0}^{2\pi}\left(-\hat{t}\ \left(\frac{p}{r}\right)^2\ \frac{3}{2}\ \left(3\ \sin^2 i\ \sin^2 u\ -\ 1\right)\ - \ \left(2\ \hat{r}-\frac{V_r}{V_t}\ \hat{t}\right)\ \left(\frac{p}{r}\right)^2\ 3\ \sin^2 i \cos u\ \sin u\right)du
| |
| </math>|{{EquationRef|30}}}}
| |
| | |
| where:
| |
| :<math>\hat{r}=\cos u\ \hat{G}\ +\ \sin u\ \hat{H}\,</math>
| |
| :<math>\hat{t}=-\sin u\ \hat{G}\ +\ \cos u\ \hat{H}\,</math>
| |
| :<math>\frac{p}{r}\ =\ 1\ +\ e_g\ \cos u\ +\ e_h\ \sin u</math>
| |
| :<math>\frac{V_r}{V_t}\ =\ \frac{e_g\ \sin u\ -\ e_h\ \cos u}{\frac{p}{r}}</math>
| |
| | |
| has the value
| |
| # {{NumBlk|:|<math>
| |
| -2\pi\ \frac{3}{2} \left(\frac{3}{2}\ \sin^2 i\ -\ 1\right)\ \left(-e_h \hat{G}\ +\ e_g \hat{H}\right)
| |
| </math>|{{EquationRef|31}}}}
| |
| | |
| Integrating the first term of the integrand one gets:
| |
| | |
| # {{NumBlk|:|<math>
| |
| \begin{align}
| |
| &\int\limits_{0}^{2\pi}-t_g\ \left(\frac{p}{r}\right)^2\ \frac{3}{2}\ 3\ \sin^2 i\ \sin^2 u\ du\ =\
| |
| \frac{9}{2}\ \sin^2 i\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_g\ \cos u\ +\ e_h\ \sin u\right)^2\ \ \sin^3 u\ du\ = \\
| |
| &9\ \sin^2 i\ e_h\ \int\limits_{0}^{2\pi}\sin^4 u\ du\ =\ 2\pi \frac{27}{8}\ \sin^2 i\ e_h
| |
| \end{align}
| |
| </math>|{{EquationRef|32}}}}
| |
| | |
| and
| |
| | |
| # {{NumBlk|:|<math>
| |
| \begin{align}
| |
| &\int\limits_{0}^{2\pi}-t_h\ \left(\frac{p}{r}\right)^2\ \frac{3}{2}\ 3\ \sin^2 i\ \sin^2 u\ du\ =\
| |
| -\frac{9}{2}\ \sin^2 i\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_g\ \cos u\ +\ e_h\ \sin u\right)^2\ \ \sin^2 u\ \cos u\ du\ = \\
| |
| &-9\ \sin^2 i\ e_g\ \int\limits_{0}^{2\pi}\sin^2 u\ \cos^2 u\ du\ =\ -2\pi \frac{9}{8}\ \sin^2 i\ e_g
| |
| \end{align}
| |
| </math>|{{EquationRef|33}}}}
| |
| | |
| For the second term one gets:
| |
| | |
| # {{NumBlk|:|<math>
| |
| \int\limits_{0}^{2\pi} t_g\ \left(\frac{p}{r}\right)^2\ \frac{3}{2}\ du\ =\
| |
| -\frac{3}{2}\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_g\ \cos u\ +\ e_h\ \sin u\right)^2\ \ \sin u\ du\ =\
| |
| -3\ e_h\ \int\limits_{0}^{2\pi}\sin^2 u\ du\ =\ -2\pi \frac{3}{2}\ e_h
| |
| </math>|{{EquationRef|34}}}}
| |
| | |
| and
| |
| | |
| # {{NumBlk|:|<math>
| |
| \int\limits_{0}^{2\pi} t_h\ \left(\frac{p}{r}\right)^2\ \frac{3}{2}\ du\ =\
| |
| \frac{3}{2}\ \int\limits_{0}^{2\pi}\ \left(1\ +\ e_g\ \cos u\ +\ e_h\ \sin u\right)^2\ \ \cos u\ du\ =\
| |
| 3\ e_g\ \int\limits_{0}^{2\pi}\cos^2 u\ du\ =\ 2\pi \frac{3}{2}\ e_g
| |
| </math>|{{EquationRef|35}}}}
| |
| | |
| For the third term one gets:
| |
| # {{NumBlk|:|<math>
| |
| \begin{align}
| |
| &-\int\limits_{0}^{2\pi}\ 2\ r_g \ \left(\frac{p}{r}\right)^2\ 3\ \sin^2 i \cos u\ \sin u\ du\ =\
| |
| -6\ \sin^2 i \int\limits_{0}^{2\pi}\ \left(1\ +\ e_g\ \cos u\ +\ e_h\ \sin u\right)^2\ \cos^2 u\ \sin u\ du\ =\ \\
| |
| &-12\ \sin^2 i\ e_h \int\limits_{0}^{2\pi}\ \cos^2 u\ \sin^2 u\ du\ =\ -2\pi \frac{3}{2}\ \sin^2 i\ e_h
| |
| \end{align}
| |
| </math>|{{EquationRef|36}}}}
| |
| | |
| and
| |
| | |
| # {{NumBlk|:|<math>
| |
| \begin{align}
| |
| &-\int\limits_{0}^{2\pi}\ 2\ r_h \ \left(\frac{p}{r}\right)^2\ 3\ \sin^2 i \cos u\ \sin u\ du\ =\
| |
| -6\ \sin^2 i \int\limits_{0}^{2\pi}\ \left(1\ +\ e_g\ \cos u\ +\ e_h\ \sin u\right)^2\ \cos u\ \sin^2 u\ du\ =\ \\
| |
| &-12\ \sin^2 i\ e_g \int\limits_{0}^{2\pi}\ \sin^2 u \cos^2 u\ du\ =\ -2\pi \ \frac{3}{2}\ \sin^2 i\ e_g
| |
| \end{align}
| |
| </math>|{{EquationRef|37}}}}
| |
| | |
| For the fourth term one gets:
| |
| | |
| # {{NumBlk|:|<math>
| |
| \begin{align}
| |
| &\int\limits_{0}^{2\pi}t_g\ \frac{V_r}{V_t}\ \left(\frac{p}{r}\right)^2\ 3\ \sin i \cos^2 u\ \sin u\ du\ =
| |
| -3\ \sin^2 i \int\limits_{0}^{2\pi}(e_g\ \sin u\ -\ e_h\ \cos u)\ \frac{p}{r}\ \cos u\ \sin^2 u\ du \ = \\
| |
| &-3\ \sin^2 i \int\limits_{0}^{2\pi}(e_g\ \sin u\ -\ e_h\ \cos u)\ (1\ +\ e_g\ \cos u\ +\ e_h\ \sin u)\ \cos u\ \sin^2 u\ du\ = \\
| |
| &3\ \sin^2 i \ e_h\int\limits_{0}^{2\pi}\ \ \cos^2 u\ \sin^2 u\ du\ =\ 2\pi \frac{3}{8} \sin^2 i \ e_h
| |
| \end{align}
| |
| </math>|{{EquationRef|38}}}}
| |
| | |
| and
| |
| | |
| # {{NumBlk|:|<math>
| |
| \begin{align}
| |
| &\int\limits_{0}^{2\pi}t_h\ \frac{V_r}{V_t}\ \left(\frac{p}{r}\right)^2\ 3\ \sin^2 i \cos u\ \sin u\ du\ =
| |
| 3\ \sin^2 i \int\limits_{0}^{2\pi}(e_g\ \sin u\ -\ e_h\ \cos u)\ \frac{p}{r}\ \cos^2 u\ \sin u\ du \ = \\
| |
| &3\ \sin^2 i \int\limits_{0}^{2\pi}(e_g\ \sin u\ -\ e_h\ \cos u)\ (1\ +\ e_g\ \cos u\ +\ e_h\ \sin u)\ \cos^2 u\ \sin u\ du\ = \\
| |
| &3\ \sin^2 i \ e_g\int\limits_{0}^{2\pi}\ \ \cos^2 u\ \sin^2 u\ du\ =\ 2\pi \frac{3}{8} \sin^2 i \ e_g
| |
| \end{align}
| |
| </math>|{{EquationRef|39}}}}
| |
| | |
| Adding the right hand sides of ({{EquationNote|32}}), ({{EquationNote|34}}), ({{EquationNote|36}}) and ({{EquationNote|38}}) one gets
| |
| <math>
| |
| 2\pi \frac{27}{8}\ \sin^2 i\ e_h\ -\ 2\pi \frac{3}{2}\ e_h\ -\ 2\pi \frac{3}{2}\ \sin^2 i\ e_h\ +\ 2\pi \frac{3}{8} \sin^2 i \ e_h
| |
| \ =\ 2\pi\ \frac{3}{2} \left(\frac{3}{2}\ \sin^2 i\ -\ 1\right)\ e_h
| |
| </math>
| |
| | |
| Adding the right hand sides of ({{EquationNote|33}}), ({{EquationNote|35}}), ({{EquationNote|37}}) and ({{EquationNote|39}}) one gets
| |
| <math>
| |
| -2\pi \frac{9}{8}\ \sin^2 i\ e_g\ +\ 2\pi \frac{3}{2}\ e_g\ -\ 2\pi \ \frac{3}{2}\ \sin^2 i\ e_g\ +\ 2\pi \frac{3}{8} \sin^2 i \ e_g\ =\ -2\pi\ \frac{3}{2} \left(\frac{3}{2}\ \sin^2 i\ -\ 1\right)\ e_g
| |
| </math>
| |
| | |
| ==References==
| |
| | |
| * El'Yasberg "Theory of flight of artificial earth satellites", '''Israel program for Scientific Translations (1967)'''
| |
| | |
| ==See also==
| |
| * [[Frozen orbit]]
| |
| * [[Molniya orbit]]
| |
| | |
| [[Category:Orbital perturbations]]
| |
| [[Category:Spaceflight concepts]]
| |