Perturbed Motion - Revision history

Claudia.Prajanu at 22:20, 21 July 2018

2018-07-21T22:20:32Z

← Older revision		Revision as of 22:20, 21 July 2018
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	:The oblateness of the Earth (<math>\displaystyle J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math>\displaystyle \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table shows the rate of change for the [[GPS]], [[BeiDou_General_Introduction\|BeiDou]] (MEO), [[~~GALILEO~~\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites. ([Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>)		:The oblateness of the Earth (<math>\displaystyle J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math>\displaystyle \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table shows the rate of change for the [[GPS]], [[BeiDou_General_Introduction\|BeiDou]] (MEO), [[Galileo General Introduction\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites. ([Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>)

Filipe.Pelica at 17:47, 26 May 2014

2014-05-26T17:47:54Z

← Older revision		Revision as of 17:47, 26 May 2014
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	:The oblateness of the Earth (<math>\displaystyle J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math>\displaystyle \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table shows the rate of change for the [[GPS]], ~~Compass~~ (MEO), [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites. ([Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>)		:The oblateness of the Earth (<math>\displaystyle J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math>\displaystyle \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table shows the rate of change for the [[GPS]], [[BeiDou_General_Introduction\|BeiDou]] (MEO), [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites. ([Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>)

Jaume.Sanz at 11:38, 13 January 2013

2013-01-13T11:38:40Z

← Older revision		Revision as of 11:38, 13 January 2013
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	:::<math>		:::<math>
	\begin{array}{ll}		\begin{array}{ll}
	V= & \frac{\mu}{r}\left[ 1- \displaystyle		V= & \displaystyle \frac{\mu}{r}\left[ 1- \displaystyle
	\sum_{n=2}^{\infty}{\left(\frac{a_e}{r}\right)^n J_n\;		\sum_{n=2}^{\infty}{\left(\frac{a_e}{r}\right)^n J_n\;
	P_n(\sin \phi)} \right .\\		P_n(\sin \phi)} \right .\\
	& + \left. \sum_{n=2}^{\infty}{\displaystyle		& + \left. \displaystyle \sum_{n=2}^{\infty}{\displaystyle
	\sum_{m=1}^{~~\infty~~}{\left(\frac{a_e}{r}\right)^n		\sum_{m=1}^{n}{\left(\frac{a_e}{r}\right)^n
	\left[ C_{nm} \cos m\lambda + S_{nm} \sin m\lambda \right ] P_{nm}(\sin		\left[ C_{nm} \cos m\lambda + S_{nm} \sin m\lambda \right ] P_{nm}(\sin
	\phi)}}\right ]		\phi)}}\right ]

Jaume.Sanz at 11:33, 13 January 2013

2013-01-13T11:33:19Z

← Older revision		Revision as of 11:33, 13 January 2013
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	:As commented before, the shape of the Earth can be approximated by an ellipsoid, with equatorial radius about <math>20\,km</math> larger than polar radius. On the other hand, the density is not uniform and the gravitational force depends on the latitude and longitude besides the radial distance. Thus, the Earth potential <ref group="footnotes">In equation <math> \mathbb{\mathbf {\ddot r}}+\displaystyle\frac{\mu}{r^3}\mathbb{\mathbf r}=\mathbb{\mathbf 0}		:As commented before, the shape of the Earth can be approximated by an ellipsoid, with equatorial radius about <math>20\,km</math> larger than polar radius. On the other hand, the density is not uniform and the gravitational force depends on the latitude and longitude besides the radial distance. Thus, the Earth potential <ref group="footnotes">In equation <math> \mathbb{\mathbf {\ddot r}}+\displaystyle\frac{\mu}{r^3}\mathbb{\mathbf r}=\mathbb{\mathbf 0}
	</math>, the term <math>\frac{\mu}{r^3}\mathbb{\mathbf r}=-\nabla (\mu/r) </math> corresponds to the acceleration from a potential produced by a spherical Earth with uniform density, see also equation (1) in [[GNSS Broadcast Orbits]].</ref> can be represented by a spherical harmonic expansion [Hofmann-Wellenhof et al., 2008] <ref>[Hofmann-Wellenhof et al., 2008] Hofmann-Wellenhof, B., Lichtenegger, H., K. and Wasle, E., 2008. GNSS - Global Navigation Satellite Systems.. Springer-Verlag, Wien, Austria </ref> like:		</math>, the term <math>-\frac{\mu}{r^3}\mathbb{\mathbf r}=\nabla (\mu/r) </math> corresponds to the acceleration from a potential produced by a spherical Earth with uniform density, see also equation (1) in [[GNSS Broadcast Orbits]].</ref> can be represented by a spherical harmonic expansion [Hofmann-Wellenhof et al., 2008] <ref>[Hofmann-Wellenhof et al., 2008] Hofmann-Wellenhof, B., Lichtenegger, H., K. and Wasle, E., 2008. GNSS - Global Navigation Satellite Systems.. Springer-Verlag, Wien, Austria </ref> like:

	:::<math>		:::<math>
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	:The oblateness of the Earth (<math>\displaystyle J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math>\displaystyle \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table (~~from~~ [Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>) ~~shows the rate of change for the [[GPS]], [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites.~~		:The oblateness of the Earth (<math>\displaystyle J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math>\displaystyle \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table shows the rate of change for the [[GPS]], Compass (MEO), [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites. ([Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>)

Jaume.Sanz at 12:20, 26 December 2012

2012-12-26T12:20:59Z

← Older revision		Revision as of 12:20, 26 December 2012
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	: '''2. ''The presence of other celestial bodies, foremost, the sun and the moon:'''''		: '''2. ''The presence of other celestial bodies, foremost, the sun and the moon:'''''
	:The gravitational field of sun and moon act as perturbing forces on the satellites, being the moon the body producing the largest effects <ref group="footnotes"> Note that, although being the sun much more massive than the moon, it is also much more farther away, and the gravitational effect is proportional to the inverse of the ~~cube~~ of the distance.</ref>. These gravitational forces also produce tides deforming the shape of the Earth and affecting its gravitational potential. Nevertheless, these tidal effects produce on GNSS satellites accelerations at the order of <math>\displaystyle 10^{-9} m/s^2</math>, which are three orders of magnitude lower than the moon and solar gravitational acceleration (see Table 3).		:The gravitational field of sun and moon act as perturbing forces on the satellites, being the moon the body producing the largest effects <ref group="footnotes"> Note that, although being the sun much more massive than the moon, it is also much more farther away, and the gravitational effect is proportional to the inverse of the square of the distance.</ref>. These gravitational forces also produce tides deforming the shape of the Earth and affecting its gravitational potential. Nevertheless, these tidal effects produce on GNSS satellites accelerations at the order of <math>\displaystyle 10^{-9} m/s^2</math>, which are three orders of magnitude lower than the moon and solar gravitational acceleration (see Table 3).

Carlos.Lopez at 11:34, 23 February 2012

2012-02-23T11:34:58Z

← Older revision		Revision as of 11:34, 23 February 2012
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	\|Level=Intermediate		\|Level=Intermediate
	\|YearOfPublication=2011		\|YearOfPublication=2011
	~~\|Logo=gAGE~~
	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}

Jaume.Sanz at 08:57, 27 January 2012

2012-01-27T08:57:48Z

← Older revision		Revision as of 08:57, 27 January 2012
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	:The oblateness of the Earth (<math>J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math> \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table (from [Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>) shows the rate of change for the [[GPS]], [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites.		:The oblateness of the Earth (<math>\displaystyle J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math>\displaystyle \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table (from [Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>) shows the rate of change for the [[GPS]], [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites.


	:::[[File:Perturbed_Motion_Table_1.PNG\|none\|thumb\|640px\|'''''Table 1:''''' Secular precession of GNSS ascending node produced by <math>J_2</math>.]]		:::[[File:Perturbed_Motion_Table_1.PNG\|none\|thumb\|640px\|'''''Table 1:''''' Secular precession of GNSS ascending node produced by <math>\displaystyle J_2</math>.]]


	:The second effect of the non-central geo-potential (<math>J_2</math>) produces a rotation of the perigee <math> \frac{d \omega}{dt}</math> (i.e., a rotation of the major axis in the orbital plane).		:The second effect of the non-central geo-potential (<math>\displaystyle J_2</math>) produces a rotation of the perigee <math>\displaystyle \frac{d \omega}{dt}</math> (i.e., a rotation of the major axis in the orbital plane).


	:::[[File:Perturbed_Motion_Table_2.PNG\|none\|thumb\|640px\|'''''Table 2:''''' Secular precession of GNSS perigee produced by <math>J_2</math>.]]		:::[[File:Perturbed_Motion_Table_2.PNG\|none\|thumb\|640px\|'''''Table 2:''''' Secular precession of GNSS perigee produced by <math>\displaystyle J_2</math>.]]


	:This second harmonic of the geopotential (<math>J_2</math>) does not produce secular perturbations in the elements <math>a</math>, <math>e</math> and <math>i</math>.		:This second harmonic of the geopotential (<math>\displaystyle J_2</math>) does not produce secular perturbations in the elements <math>\displaystyle a</math>, <math>\displaystyle e</math> and <math>\displaystyle i</math>.



	: '''2. ''The presence of other celestial bodies, foremost, the sun and the moon:'''''		: '''2. ''The presence of other celestial bodies, foremost, the sun and the moon:'''''
	:The gravitational field of sun and moon act as perturbing forces on the satellites, being the moon the body producing the largest effects <ref group="footnotes"> Note that, although being the sun much more massive than the moon, it is also much more farther away, and the gravitational effect is proportional to the inverse of the cube of the distance.</ref>. These gravitational forces also produce tides deforming the shape of the Earth and affecting its gravitational potential. Nevertheless, these tidal effects produce on GNSS satellites accelerations at the order of <math>10^{-9} m/s^2</math>, which are three orders of magnitude lower than the moon and solar gravitational acceleration (see Table 3).		:The gravitational field of sun and moon act as perturbing forces on the satellites, being the moon the body producing the largest effects <ref group="footnotes"> Note that, although being the sun much more massive than the moon, it is also much more farther away, and the gravitational effect is proportional to the inverse of the cube of the distance.</ref>. These gravitational forces also produce tides deforming the shape of the Earth and affecting its gravitational potential. Nevertheless, these tidal effects produce on GNSS satellites accelerations at the order of <math>\displaystyle 10^{-9} m/s^2</math>, which are three orders of magnitude lower than the moon and solar gravitational acceleration (see Table 3).

Jaume.Sanz at 08:54, 27 January 2012

2012-01-27T08:54:06Z

← Older revision		Revision as of 08:54, 27 January 2012
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	:The oblateness of the Earth (<math>J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math> \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table (from [Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135{150.</ref>) shows the rate of change for the [[GPS]], [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites.		:The oblateness of the Earth (<math>J_2</math>) produces two effects: First, a torque which rotates the satellite's orbit in the equatorial plane, producing a nodal regression <math> \frac{d \Omega}{dt}</math>. This is a secular effect (i.e., timely cumulative) which depends upon the inclination of the orbit. It is zero for a polar orbit and maximum for an equatorial orbit. Next table (from [Cojocaru, 2009] <ref> [Cojocaru, 2009] Cojocaru, S., 2009. GPS-GLONASS-GALILEO: A Dynamical Comparison. The Journal of Navigation, The Royal Institute of Navigation, UK. 62, pp. 135-150.</ref>) shows the rate of change for the [[GPS]], [[GALILEO\|Galileo]] and [[GLONASS General Introduction\|GLONASS]] satellites.

Jaume.Sanz at 08:41, 27 January 2012

2012-01-27T08:41:46Z

← Older revision		Revision as of 08:41, 27 January 2012
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	:As commented before, the shape of the Earth can be approximated by an ellipsoid, with equatorial radius about <math>20\,km</math> larger than polar radius. On the other hand, the density is not uniform and the gravitational force depends on the latitude and longitude besides the radial distance. Thus, the Earth potential <ref group="footnotes">In equation <math> \mathbb{\mathbf {\ddot r}}+\displaystyle\frac{\mu}{r^3}\mathbb{\mathbf r}=\mathbb{\mathbf 0}		:As commented before, the shape of the Earth can be approximated by an ellipsoid, with equatorial radius about <math>20\,km</math> larger than polar radius. On the other hand, the density is not uniform and the gravitational force depends on the latitude and longitude besides the radial distance. Thus, the Earth potential <ref group="footnotes">In equation <math> \mathbb{\mathbf {\ddot r}}+\displaystyle\frac{\mu}{r^3}\mathbb{\mathbf r}=\mathbb{\mathbf 0}
	</math>, the term <math>\frac{\mu}{r^3}\mathbb{\mathbf r}=-\nabla (\mu/r) </math> corresponds to the acceleration from a potential produced by a spherical Earth with uniform density.</ref> can be represented by a spherical harmonic expansion [Hofmann-Wellenhof et al., 2008] <ref>[Hofmann-Wellenhof et al., 2008] Hofmann-Wellenhof, B., Lichtenegger, H., K. and Wasle, E., 2008. GNSS - Global Navigation Satellite Systems.. Springer-Verlag, Wien, Austria </ref> like:		</math>, the term <math>\frac{\mu}{r^3}\mathbb{\mathbf r}=-\nabla (\mu/r) </math> corresponds to the acceleration from a potential produced by a spherical Earth with uniform density, see also equation (1) in [[GNSS Broadcast Orbits]].</ref> can be represented by a spherical harmonic expansion [Hofmann-Wellenhof et al., 2008] <ref>[Hofmann-Wellenhof et al., 2008] Hofmann-Wellenhof, B., Lichtenegger, H., K. and Wasle, E., 2008. GNSS - Global Navigation Satellite Systems.. Springer-Verlag, Wien, Austria </ref> like:

	:::<math>		:::<math>

Jaume.Sanz at 08:35, 27 January 2012

2012-01-27T08:35:06Z

← Older revision		Revision as of 08:35, 27 January 2012
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	:where <math>\displaystyle a_e</math> is the semi-major axis of the Earth, <math>\displaystyle r</math> is the geocentric distance of the satellite, and <math>\displaystyle \phi</math>, <math>\displaystyle \lambda</math> are its latitude and longitude. The parameters <math>\displaystyle J_n</math>, <math>\displaystyle C_{nm}</math>, <math>\displaystyle S_{nm}</math> denote the zonal (<math>\displaystyle m=0</math>) and tesseral (<math>\displaystyle m \neq 0\ </math>) coefficients of the harmonic development known from an Earth model. <math>\displaystyle P_n</math> and <math>\displaystyle P_{nm}</math> are the Legendre polynomials and the associated Legendre functions, respectively.		:where <math>\displaystyle a_e</math> is the semi-major axis of the Earth, <math>\displaystyle r</math> is the geocentric distance of the satellite, and <math>\displaystyle \phi</math>, <math>\displaystyle \lambda</math> are its latitude and longitude. The parameters <math>\displaystyle J_n</math>, <math>\displaystyle C_{nm}</math>, <math>\displaystyle S_{nm}</math> denote the zonal (<math>\displaystyle m=0</math>) and tesseral (<math>\displaystyle m \neq 0</math> ) coefficients of the harmonic development known from an Earth model. <math>\displaystyle P_n</math> and <math>\displaystyle P_{nm}</math> are the Legendre polynomials and the associated Legendre functions, respectively.