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'''Nodal precession''' is the precession of an orbital plane around the rotation axis of an astronomical body such as the Earth. This precession is due to the non-spherical nature of a spinning body, which creates a non-uniform gravitational field.
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Around a spherical body, an orbital plane would remain fixed in space around the central body. However, most bodies rotate, which causes an equatorial bulge. This bulge creates a gravitational effect that causes orbits to [[precess]] around the rotational axis of the central body.
 
The direction of precession is opposite the direction of revolution. For a typical prograde (in the direction of central body rotation) satellite orbit in an about the Earth, the [[longitude of the ascending node]] decreases, i.e. node precesses westward. If the orbit is retrograde, this increases the longitude of the ascending node i.e. node precesses eastward. This nodal progression enables  [[sun-synchronous orbit]]s to maintain approximately constant [[Beta angle|angle relative to the Sun]].
 
==Description==
 
[[File:Nodal precession2.png|thumb|Equatorial bulge torques a satellite orbit, leading to nodal precession]]
 
A non-rotating body of planetary scale or larger would be pulled by gravity into a sphere. Virtually all bodies rotate, however. The centrifugal force deforms the body so that it has an [[equatorial bulge]]. Because of the bulge the gravitational force on the satellite is not directly toward the center of the central body, but is offset toward the equator. Whichever hemisphere the satellite is in it is preferentially pulled slightly toward the equator. This creates a torque on the orbit. This torque does not reduce the inclination; rather, it causes a torque-induced gyroscopic [[precession]], which causes the [[orbital nodes]] to drift with time.
 
==Equation==
 
The rate of precession depends on the [[inclination]] of the orbital plane to the equatorial plane, as well as the orbital eccentricity.
 
For a satellite in a [[prograde orbit]] about the Earth, the precession is westward (nodal regression), the node and satellite move in opposite directions.<ref>{{cite book|last=Brown|first=Charles|title=Elements of spacecraft design|pages=106|url=http://books.google.com/books?id=mTSSMhcmVbkC&lpg=PA95&ots=hkARYnIy9h&dq=ascending%20node%20regression&pg=PA121#v=onepage&q=ascending%20node%20regression&f=false}}</ref> A good approximation of the precession rate is
 
: <math>\omega_p = -\frac{3}{2}\frac{R_E^2}{(a(1-e^2))^2} J_2 \omega \cos i</math>
 
where
:<math>\omega_p \,</math> is the precession rate ([[radian|rad]]/s)
:<math>R_E \,</math> is the body's equatorial radius (6 378 137 m for Earth)
:<math>a \,</math> is the semi-major axis of the satellite orbit
:<math>e \,</math> is the eccentricity of the satellite orbit
:<math>\omega \,</math> is its angular frequency of the satellite's motion (<math>2\pi</math> radians divided by its period)
:<math>i \,</math> its inclination
:<math>J_2 \,</math> is the body's ''second dynamic form factor'' (<math> \scriptstyle -\sqrt{5}C_{20}=1.08262668\times10^{-3}</math> for Earth.)
 
This last quantity is related to the [[oblateness]] as follows:
 
: <math>J_2 = \frac{2 \varepsilon_E}{3} - \frac{R_E^3 \omega_E^2}{3 G M_E}</math>
 
where
:<math>\varepsilon_E \,</math> is the central body's oblateness
:<math>R_E \,</math> is central body's equatorial radius
:<math>\omega_E \,</math> is the central body's rotation rate (7.292115&times;10<sup>&minus;5</sup> rad/s for Earth)
:<math>G M_E \,</math> is the product of the universal [[gravitational constant|constant of gravitation]] and the central body's mass (3.986004418&times;10<sup>14</sup> m<sup>3</sup>/s<sup>2</sup> for Earth.)
 
The nodal progression of low earth orbits is typically a few degrees per day to the west (negative). For a satellite in a circular 800&nbsp;km altitude orbit at 56° inclination about the earth
 
: <math>R_E = 6.378 137 \times10^{6}m</math>
: <math>J_2 = 1.08262668\times10^{-3}</math>
 
The orbital period is 6052.4 s, so the angular velocity is 0.001038 rad/s. The precession is therefore
 
:<math>\omega_p = -\frac{3}{2}\frac{6 378 137^2}{(7 178 137(1-0^2))^2}\ 1.08262668\times10^{-3}\ 0.001038\ \cos 56^{\circ}</math>
 
:<math>\omega_p = -7.44\times10^{-7}\ rad/s</math>
 
This is equivalent to -3.683 degrees/day, so the orbit plane will make one complete turn (in inertial space) in 98 days.
 
The apparent motion of the sun is approximately +1° per day (360° per year / 365.2422 days per tropical year ≈ 0.9856473° per day), so apparent motion of the sun relative to the orbit plane is about 2.8° per day, resulting in a complete cycle in about 127 days. For retrograde orbits <math>\omega</math> is negative, so the precession becomes positive. (Alternatively, <math>\omega</math> can be thought of as positive but the inclination is greater than 90°, so the cosine of the inclination is negative.) In this case it is possible to make the precession approximately match the apparent motion of the sun, resulting in a [[sun-synchronous orbit]].
 
==See also==
* [[Axial precession (astronomy)|Axial precession]] or precession of the equinoxes in astronomy (same kind)
* [[Apsidal precession]], the other kind of precession ([[argument of periapsis]] change)
 
==References==
{{Reflist}}
 
==External links==
*[http://yarchive.net/space/orbits/nodal_regression.html Nodal regression description from USENET]
*[http://www.agi.com/resources/academic-resources/astro-primer/primer91.htm Discussion of nodal regression from Analytical Graphics]
 
[[Category:Astrodynamics]]

Revision as of 11:43, 5 March 2014

While this mindset is certainly understandable, it simply isn't practical at that price point. Various means of transportation come second - from yachts or boats and cars to motorcycles. The downhill bikes have both front and rear suspension, disc breaks and are very strong. If they aren't a know brand, don't panic, I'm certain they'll be fit for purpose, but bear in mind if they look really cheap then you may struggle with things such as adjustment of your gears. If you've ridden for years and years and know what you need, buying online does make sense.

Similarly mountain bike wheels are also an important feature. Young children can hardly wait for their first Hot Wheels or tricycle. If you don't do a lot of racing on mountain trails then you need a road bike. Just to give you an idea, mine has three settings, those being; *Completely off, so that I am riding it as a normal bike; *Power assist, where the power kicks in after a couple of revolutions of the pedals and then remains on as long as the pedals are turning, (there is a high and low setting in this mode). Browse a number of the on-line forums to determine what other riders like and do not like about their bicycles.

Given below is a list of essential items that you should carry along whenever you set off on a this ride. The big three has a major part to play on this matter and so they should step up to be strong and mature leaders. If you liked this post and you would such as to receive even more details relating to Here is your mountain bike sizing. kindly browse through the internet site. There are great coaches, who will train you to tweet like a bird in spring or Link in, befriend and get a fan page. Mounting biking is a fun sport for professionals and beginners alike. If you happen to crash, then you will see that your hands are the first thing that will probably come into contact with the ground.

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