Relativity for Global Navigation Satellite Systems

Notes by Anna Heffernan based on the Living reviews article, “Relativity in the Global Positioning Systems”, Neil Ashby, Living Rev. Relativity 6, (2003),1 which will be referred to as [1] in the below notes.  A good accompaniment is Section 10.3 of Gravity by Eric Poisson and Clifford Will.  This notebook serves as a supplement to [1] hence equations and sections have been labelled accordingly - it in no means replaces the more complete material provided in [1].  Interested students can avail themselves of a more thorough understanding by reading the Living reveiws article (as well as Gravity S10.3).

S1 Introduction

The system

3 segments

GPS has 3 principle segments:
		- Space segment
		- Control segment
		- User segment

Space segment
	- 24-32 atomic clock carrying satellites (GPS:24 but spares exist)
	- organised in several tilted orbital planes (GPS: 4 satellites in 6 planes inclines at 55° to earths’ equator)
	- timing signals calculated from clocks are transmitted from each satellite,
	- i.e., event: satellites emit signal with spacetime coordinates of the emission.
	- additional information includes information on: the entire satellite constellation, satellite health and universal coodinated time

Control segment
	- ground-based monitoring stations (keep track of information from the satellites)
	- satellite information is analyized and projects the natural positions and clock behaviour of the satellites for next few hours
	- this is uploaded to the satellites for retransmission to users.

User segment
	- receive signals transmitted from the satellites and use them to determine position, velocity and time on their local clocks.

Technology

The main techonology for GPS is the extremely accurate, stable clocks.  Fig. 1 (taken from [1]) plots the Allen deviation, for both a high-performance Cessium clock and a Quartz oscillator (typical for GPS receiver), as a function of sample time τ.   We can see that Quartz oscillators have better short term stability than Cesium clocks but after 100 seconds or so, Cesium has far better performance.  FIg. 1 is for guidance - in reality there is variation aroundt the nominal values.

The plot for Cesium characterizes the best orbiting clocks in the GPS system => after initializing a Cesium clock, and leaving it alone for a day (~(10^6)s), it should be correct to within 5 parts in (10^14) or 4 ns.  Relativistic effects are huge compared to this.

Relativistic Effects

Despite a weak gravitational field and small clock velocities, significant relativistic effects are present in the GPS system, these include,
	- first and second order Doppler frequency shifts of clocks due to their relative motion
	- gravitational frequency shifts
	- Sagnac effect due to Earth’s rotation.
If unaccounted for => large errors in GPS navigation and time transfer.

To determine position and time transfer with GPS, one considers 4 synchronized atomic clocks transmitting their time and position ((Subscript[t, j]),(Subscript[(OverVector[r]), j])) for j = 1,2,3,4.  These four signals are received at position (t,(OverVector[r]))
		=>(((c^2)(t-(Subscript[t, j])))^2)=|(OverVector[r])-(Subscript[(OverVector[r]), j])((|)^2), 		j = 1,2,3,4				;(1)
where c is the speed of light 	=> 4 unknowns (t,(Subscript[(OverVector[r]), j])) with 4 equations 
					=> solvable.
					=>Timing errors of 1 ns => 30 cm position error (c~3×(10^8), Δ t~(10^(-9))).  
We will see however that one must be very careful in specifying the reference frame in which the transmitter clocks are synchronized to ensure (1) remains valid.

S2 Reference Frames and the Sagnac Effect

Reference Frames

ECEF: Earth-Centered, Earth-Fixed reference frame

- Earth-Centered, Eart-fixed in which Earth rotates around a fixed axis with a defined rotation, (Subscript[ω, E]).
- Earth actually rotates about a slightly different axis with variable rotation but this is of little consequence for relativity
- As GPS users are fixed or slowly moving on the Earth’s surface => early decision to describe satellite’s position in ECEF
- ECEF satellite position => user position will be outputted in ECEF also
- Rotating => not inertial =>to use (1) we need a different (inertial) reference frame

Tranforming from inertial to rotating frames

Consider transformation from an inertial frame, in which spacetime is MInkowskian, to a rotating frame of reference (ignoring gravitational potential for now), the metric in an inertial frame in cylindrical coordinates is,
				-d (s^2)=-(c^2) d (t^2)+d (r^2)+(r^2)((dϕ)^2)+d (z^2),					;(2)
and the transformation is given by,
			t = t’, 		r = r’, 		ϕ=ϕ'+(Subscript[ω, E])t',		z = z’				;(3)
			=>d (s^2)=-(1-((((Subscript[ω, E])^2) ((r')^2))/(c^2)))(c^2)d ((t')^2)+2 (Subscript[ω, E]) ((r')^2) d ϕ' d t'+d ((σ')^2),				;(4)
where			d ((σ')^2)=d ((r')^2)+((r')^2)d ((ϕ')^2)+d ((z')^2).
and t is coordinate time => proper time as measured by an inertial observer at infinity.

Synchronization

Sagnac Effect

To solve (1) for the user’s position, we must be able to compare and synchronize times, one can use light to translate this information as the speed of light is constant in every frame. Another option is a slowly portable clock.  Regardless of the method, (1) will not hold in a rotating frame due to the Sagnac effect => we must carry out our calculations in a local inertial frame or ECI coordinate system.

Synchronization via light

A light ray is emitted at A and received at B: Light => (ds^2)=0 in (4), and keeping only terms of first order in (((Subscript[ω, E]) r')/c), (4) becomes,
			(c^2)((dt')^2)-((dσ')^2)-2  ((r')^2)dϕ' c dt'(((Subscript[ω, E])/c))+((O(((Subscript[ω, E])/c)))^2)=0,					;(5)
			=>c dt'=dσ' [1+(((r^('2))dϕ' c dt')/((dσ')^2))(((Subscript[ω, E])/c))+((O(((Subscript[ω, E])/c)))^2)],
			=>cdt'=((dσ'[1-((((r')^2)dϕ' )/(dσ'))(((Subscript[ω, E])/c))])^(-1))+((O(((Subscript[ω, E])/c)))^2),
				=dσ'+((r')^2)dϕ' (((Subscript[ω, E])/c))+((O(((Subscript[ω, E])/c)))^2).						;(6)

The quantity ((((r')^2) dϕ')/2)=π ((r')^2) (((dϕ')/(2 π))) is simply the infinitissimal area (Subscript[dA', z]) as illustrated in Fig. 2.  The total time requied for light to travel between these two points is 		(Overscript[(∫_A), B])dt'=(1/c)(Overscript[(∫_A), B])dσ'+((2 (Subscript[ω, E]))/(c^2)) (Overscript[(∫_A), B])d (Subscript[A', z]).							;(7)
=> Observers fixed on the earth, who were unaware of the Earth’s rotation, would just use (1/c)∫dσ' to synchronize their clock network.  
=> Observers at rest in the underling inertial frame would see that this leads to significant path-dependent inconsistencies.  
=> From the underlying inertial frame view: the additional travel time required by light to catch up to the moving reference point.

Synchronisation via a portable clock

A clock slowly moves from point A to point B: A timelike geodesic => (ds^2)=-(c^2)((dτ)^2) where τ is the proper time (measured by a clock), (4) now
		=>((dτ)^2)=((dt')^2) [1-((((Subscript[ω, E])^2) ((r')^2))/(c^2))-((2 (Subscript[ω, E]) ((r')^2) d ϕ')/((c^2)dt'))-(1/(c^2))(((((dσ')/(dt'))))^2)].						;(9)
A slowly moving clock => (((((dσ')/(dt'))))^2)<<1, allowing us to disregard this term, also keeping only first order of ((((Subscript[ω, E]) r')/c)) yields,
		dτ=dt'[1-(( (Subscript[ω, E]) ((r')^2) d ϕ')/((c^2)dt'))]=dt'-(( (Subscript[ω, E]) ((r')^2) d ϕ')/(c^2)),							;(10)
		=>(Overscript[(∫_A), B])dt'=(Overscript[(∫_A), B])dτ+((2 (Subscript[ω, E]))/(c^2)) (Overscript[(∫_A), B])d (Subscript[A', z]).									;(11)
=> the time difference recorded on the clock (dτ) is different to the coordinate time difference (dt) between events A (clock departed) and B (clock arrived).  
=> Comparing with (7), regardless of whether we use light or portable clocks to disseminate time, path-dependent discrepancies in the rotating frame are inescapable => must use underlying inertial frame for synchronization.

S3 GPS Coordinate Time and TAI

Side note: Legendre polynomial generating functions

By a Taylor expansion, provided u<<1, we have
			(((1-u))^(-1/2))=(Overscript[(∑_(n=0)), ∞])(((f^((n)))(1))/(n!))(((-u))^n)
		=(Overscript[(∑_(n=0)), ∞])(((2 n)!)/((((2^n)n!))^2))(u^n),	where		f(x)=(x^(-1/2))=>(f^((n)))(x)=((((-1))^n)/(2^n)) (2n-1)!!(x^((-(2n+1))/2))
		=>(f^((n)))(1)=((((-1))^n)/(2^n)) (2n-1)!!
			    = ((((-1))^n)/(2^n)) (((2n)!)/((2n)!!))= ((((-1))^n)/(2^n))(((2n)!)/((2^n)n!)).
Now if we consider, u=2xt-(t^2), we have,
			(((1-2x t+(t^2)))^(-1/2))=(Overscript[(∑_(n=0)), ∞])(((2 n)!)/((((2^n)n!))^2))(((2xt-(t^2)))^n)
			=(Overscript[(∑_(n=0)), ∞])(((2 n)!)/((((2^n)n!))^2))(Overscript[(∑_(k=0)), n])(({{n}, {k}})) (((2 x t))^(n-k)) (((-(t^2)))^k)
			=(Overscript[(∑_(n=0)), ∞])(((2 n)!)/((((2^n)n!))^2))(Overscript[(∑_(k=0)), n])((n!)/(k! (n-k)!))(((-1))^k) (2^(n-k)) (x^(n-k))(t^(n+k))
			=(Overscript[(∑_(s=0)), ∞])(Overscript[(∑_(k=0)), [s/2]])(((2 s-2 k)!)/(([(2^(s-k))(s-k)!])^2))(((s-k)!)/(k! (s-2 k)!))(((-1))^k) (2^(s-2k)) (x^(s-2 k))(t^s)
			=(Overscript[(∑_(s=0)), ∞]) ((t^s)/((2^s)s!)) (Overscript[(∑_(k=0)), [s/2]])((s!)/(k!(s-k)!))(((-1))^k)((d^s)/(dx^s))((x^(2(s-k)))),
			=(Overscript[(∑_(s=0)), ∞]) ((t^s)/((2^s)s!)) ((d^s)/(dx^s))((((x^2)-1))^s),
			=(Overscript[(∑_(s=0)), ∞]) (t^s)(Subscript[P, s])(x), where s=n+k∴n-k≥0=>k≤(s/2)

Coordinate time

Newtonian gravitational potential multipole moment decomposition

The Newtonian gravitational potential for a point mass is given by, U=((G M)/r).  If we are to apply this to a mass distribution, we get
		U(t,(OverVector[x]))=G(∑_i)((Subscript[m, i])/(|(OverVector[x])-(Subscript[(OverVector[x]), i])|)),								where	external field =>r >r’
		  	=G∫((ρ(t,(OverVector[x'])))/(|(OverVector[x])-(OverVector[x'])|)) (d^3)x',								(OverVector[x])·(OverVector[x])'=r r' Cosγ,
		   	=G∫(((1/r)[1-2(((r')/r))Cos γ+(((((r')/r)))^2)])^(-1/2))ρ(t,(OverVector[x']))(d^3)x',				(Subscript[P, l])(Cosγ)=((4 π)/(2 l+1))(Overscript[(∑_(m=-l)), l]) (Subscript[(SuperStar[Y]), lm])(θ',ϕ') (Subscript[Y, lm])(θ,ϕ),
		 	=G∫(Overscript[(∑_(n=0)), ∞])((((r')^n)/(r^(n+1))))(Subscript[P, n])(Cosθ)ρ(t,(OverVector[x']))(d^3)x',					Cosγ=Cosθ Cosθ'+Sinθ Sinθ' Cos(ϕ-ϕ'),
		 	=((G M)/r)+(G/r)(Overscript[(∑_(n=1)), ∞])((4π)/(2 n+1))∫(((((r')/r)))^n)(Overscript[(∑_(m=-n)), n]) (Subscript[(SuperStar[Y]), nm])(θ',ϕ') (Subscript[Y, nm])(θ,ϕ)ρ(t,(OverVector[x']))(d^3)x',		(Subscript[P, 0])=1, (Subscript[I, 10])=(Subscript[I, 1-1])=(Subscript[I, 11])=0
		 	=((G M)/r)+(Overscript[(∑_(n=2)), ∞])(G/(r^(n+1)))((4π)/(2 n+1))(Overscript[(∑_(m=-n)), n])  (Subscript[I, nm])(Subscript[Y, nm])(θ,ϕ),				(Subscript[I, lm])(t)=∫ρ(t,(OverVector[x])') ((r')^l)(Subscript[(SuperStar[Y]), lm])(θ',ϕ')(d^3)x',
where (Subscript[I, lm]) are the multipole moments of the Earth’s mass distribution and we have used the fact that the dipole moments (l=1) vanish when we place the origin of the coordinate system at the body’s centre of mass => ∫ρ(OverVector[x])(d^3)x=0.  Given (Subscript[(SuperStar[Y]), 10])=(Sqrt[(3/(4 π))])Cosθ  and (Subscript[(SuperStar[Y]), 1±1])=∓(Sqrt[(3/(8 π))])Sinθ (e^(±iθ)), this can be seen by,
			(Subscript[I, 10])=(Sqrt[(3/(4π))])∫ρ(t,(OverVector[x])') ((r')^l) Cosθ (d^3)x'=(Sqrt[(3/(4π))])∫ρ(t,(OverVector[x])')z (d^3)x'=0,
			(Subscript[I, 1±1])=∓(Sqrt[(3/(8 π))])∫ρ(t,(OverVector[x])') Sinθ (e^(±iθ))(d^3)x'=∓(Sqrt[(3/(8 π))])∫ρ(t,(OverVector[x])')  (x±i y)(d^3)x'=0.
For an axially symmetric body (mass density invariant under a rotation about a symmetry axis), Taking the z-direction to be aligned with the symmetry axis, the mass density is independent of the azimuthal angle ϕ => the only non-vanishing multipole moments are m=0.
			=>U(t,(OverVector[x]))=((G M)/r)+(Overscript[(∑_(n=2)), ∞])(G/(r^(n+1))) ((4π)/(2 n+1))(Subscript[I, n0])(Subscript[Y, n0])(θ,ϕ),
		=((G M)/r)-(Overscript[(∑_(n=2)), ∞])G M(Sqrt[((4π)/(2 n+1))])((R^n)/(r^(n+1)))(Subscript[J, n]) (Subscript[Y, n0])(θ,ϕ),
		=((G M)/r)[1-(Overscript[(∑_(n=2)), ∞])((R^n)/(r^n))(Subscript[J, n]) (Subscript[P, n])(Cosθ)],			using 		(Subscript[J, l])=-(Sqrt[((4 π)/(2 l+1))])((Subscript[I, l0])/(M (R^l))),
(Subscript[P, l])(Cosθ)=(Sqrt[((4 π)/(2 l+1))])(Subscript[Y, l0])(θ),
where the characteristic length scale R is chosen to be the body’s equatorial radius and (Subscript[J, 2]) is the Earth’s quadrupole moment coefficient.  For our current purpose, keeping  up to n=2 will be sufficient, higher multipole moment contributions have a very small effect for relativity in GPS,
			=>U(t,(OverVector[x]))=((G M)/r)[1-((((R/r)))^2)(Subscript[J, 2]) (Subscript[P, 2])(Cosθ)].								(13)

Metric in Isotropic Coordinates

In istropic coordinates (r->((r'[1+(m/(2 r'))])^2)), the Schwarzschild metric transforms to,
			(ds^2)=-(((((1-(m/(2r)))/(1+(m/(2 r))))))^2)(c^2)(dt^2)+(((1+(m/(2 r))))^4)((dr^2)+(r^2)((dΩ)^2))
	->-[1-((2 m)/r)+(1/2)(((((2 m)/r)))^2)+...](c^2) (dt^2)+[1+((2 m)/r)+(3/8)(((((2 m)/r)))^2)+...]((dr^2)+(r^2)((dΩ)^2)),
where ((dΩ)^2)=((dθ)^2)+(sin^2)θ ((dϕ)^2).  Therefore to write an approximate solution of Einstein’s field equations in istropic coordinates with the assumptions of a static mass distribution and a local inertial, non-rotating, freely falling coordinate system with the origin at the Earth’s center of mass, we have,
			(ds^2)=-(1-((2 U)/(c^2))) (c^2)(dt^2)+(1+((2 U)/(c^2))) ((dr^2)+(r^2)((dΩ)^2)),							(12)
where U is the Newtonian gravitational potential.

Rotating into ECEF coordinate system

To transfer this (12) to a rotating, ECEF coordinate system via spherical coordinates, we have,
			t=t',r=r',θ=θ',ϕ=ϕ'+(Subscript[ω, E]) t'									(14)
			=>(ds^2)=-[1-((2 U)/(c^2))-((((((Subscript[ω, E]) r' Sinθ')/c)))^2)] (c^2)((dt')^2)+2 ((Subscript[ω, E])/c)((r')^2)(Sin^2)θ dϕ' c dt'+(1+((2 U)/(c^2))) ((dr^2)+(r^2)((dΩ)^2)),	(15)
where we have disgarded terms of order (1/(c^3)) and higher.  The metric coefficient (Subscript[g', 00]) in the rotating frame is,
			(Subscript[g', 00])=-[1-((2 U)/(c^2))-((((((Subscript[ω, E]) r' Sinθ')/c)))^2)] ≡-(1-((2 Φ)/(c^2))),								(16)
where Φ is the effective gravitational potential in the rotating frame, which includes the static gravitational potential of the Earth, and a centripetal potential term.

The Earth’s Geoid

TAI: Temps Atomique International

- The Earth’s geoid is defined as the surface of constant gravitational equipotential at mean sea level in ECEF.
- TAI is the primary international standard for atomic time - defined on the Earth’s geoid.
- In (12) and (15), the rate of coordinate time is determined by atomic clocks at rest at infinity - not possible in reality
- We can relate this coordinate time to that recorded on the Earth’s geoid.

Clocks on the Earth’s geoid

For a clock at rest on Earth, (15) reduces to,
			(ds^2)=-[1-((2 U)/(c^2))-((((((Subscript[ω, E]) r' Sinθ')/c)))^2)] (c^2)((dt')^2)=-(1-((2 Φ)/(c^2)))(c^2)((dt')^2).									(17)
This determines the radius of the model geoid as a function of polar angle θ’.  The numerical value of (Subscript[Φ, 0]) can be deterined at the equator where θ'=(π/2) and r’=R,
			=>((Subscript[Φ, 0])/(c^2))= ((G M)/(R (c^2)))[1-(Subscript[J, 2]) (Subscript[P, 2])(0)]+((((Subscript[ω, E])^2)(R^2))/(2 (c^2))),		where			(Subscript[P, 2])(0)=-(1/2), G M=3.986004418×(10^14)(m^3) (s^(-2)),
				=((G M)/(R (c^2)))[1+((Subscript[J, 2])/2)]+((((Subscript[ω, E])^2)(R^2))/(2 (c^2))),					(Subscript[J, 2])=1.0826300×(10^(-3)), R=6.3781370×(10^6)
				=6.95348×(10^(-10))+3.764×(10^(-13))+1.203×(10^(-12)).		(Subscript[ω, E])=7.2921151467×(10^(-5))rad (s^(-1))	
				=6.96927×(10^(-10)).											(18)
=> three distinct contributions to this effective potential:
		- 1/r contribution due to the Earth’s mass,
		- a more complicated contribution from the quadrupole potential,
		- a centripetal term due to the Earth’s rotation.
From (17) and (ds^2)=-(c^2) ((dτ)^2), for clocks on the geoid we have,
			dτ=dt'(1-((Subscript[Φ, 0])/(c^2))).												(19)
=> clocks at rest on the geoid run slow compared to clocks at rest at infinity by about 7×(10^(-10)).  
=>These effects are about (10^5) times larger than the fractional frequency stability of a high-performance Cesium clock.  

The shape of the geoid in this model can be obtained by solving (16) with (Subscript[Φ, 0]), i.e.,
			(Subscript[Φ, 0])= ((G M)/(r'))[1-((((R/(r'))))^2)((Subscript[J, 2])/2)(3 (Cos^2)θ'-1)]+((((Subscript[ω, E])^2)(r^2) (Sin^2)θ')/(2 )),
to obtain r’=r’(Cosθ’).  If we assume r’=R(1+Δr) where Δr is small, we have,
			(Subscript[Φ, 0])=((G M)/R)(1-Δr) [1-((Subscript[J, 2])/2) (1-2 Δr)(3 (Cos^2)θ'-1)]+((((Subscript[ω, E])^2)(R^2) (1+2 Δr) (1-(Cos^2)θ'))/(2 )),
				=((G M)/R) [1-Δr-((Subscript[J, 2])/2) (1-2 Δr)(3 (Cos^2)θ'-1)+((Subscript[J, 2])/2) Δr (3 (Cos^2)θ'-1)+((((Subscript[ω, E])^2)(R^3) (1+2 Δr) (1-(Cos^2)θ'))/(2 G M))],
				=((G M)/R) [1-Δr-((Subscript[J, 2])/2) (3 (Cos^2)θ'-1)+((((Subscript[ω, E])^2)(R^3) (1-(Cos^2)θ'))/(2 G M))],		as (Subscript[J, 2])<<1,((((Subscript[ω, E])^2)(R^3))/(G M))<<1
				=((G M)/R) [1+((Subscript[J, 2])/2)+((((Subscript[ω, E])^2)(R^3))/(2 G M))-Δr-(((3 (Subscript[J, 2]))/2) +((((Subscript[ω, E])^2)(R^3) )/(2 G M))) (Cos^2)θ'],
				=(Subscript[Φ, 0])-((G M)/R)[Δr+(((3 (Subscript[J, 2]))/2) +((((Subscript[ω, E])^2)(R^3) )/(2 G M))) (Cos^2)θ']
			=>Δr=-(((3 (Subscript[J, 2]))/2)+((((Subscript[ω, E])^2)(R^3) )/(2 G M)))(Cos^2)θ'.										(20)

This treatment of the gravitational field of the oblate Earth is limited by the simple model of the gavitational field.  Here, we have estimated the shape of the so-called “reference ellipsoid”, from which the actual geoid is conventionally measured.

Geoid clocks as reference clocks

In (19), (Subscript[Φ, 0]) is a constant and dτ represents the increment of proper time as recorded by a clock at rest.
=> ideal clocks at rest on the geoid of the rotating Earth all tick at the same rate.

In (12), the rate of coordiante time is defined by standard clocks at infinity, to switch to clocks on the geoid, we define a new coordinate time, t’’:
			t''=(1-((Subscript[Φ, 0])/(c^2)))t'=(1-((Subscript[Φ, 0])/(c^2)))t.		or 	t=t'=(1+((Subscript[Φ, 0])/(c^2)))t''+O((c^(-4)))					(21)
If we recall (15)	(ds^2)=-[1-((2 Φ)/(c^2))] (c^2)((dt')^2)+2 ((Subscript[ω, E])/c)((r')^2)(Sin^2)θ' dϕ' c dt'+(1+((2 U)/(c^2))) (((dr')^2)+((r')^2)((dΩ')^2)),				(15)
we now have			=-[1-((2 (Φ-(Subscript[Φ, 0])))/(c^2))] (c^2)((dt'')^2)+2 ((Subscript[ω, E])/c)((r')^2)(Sin^2)θ dϕ'' c dt''+(1+((2 U)/(c^2))) (((dr'')^2)+((r'')^2)((dΩ'')^2)),			(22)
where we have again only retained an accuracy to (c^(-2)).  Carrying out the same time scale change in the non-rotating metric (12), gives
			(ds^2)=-(1-((2 (U-(Subscript[Φ, 0])))/(c^2))) (c^2)((dt'')^2)+(1+((2 U)/(c^2))) (((dr'')^2)+((r'')^2)((dΩ'')^2)).							(23)
(22) and (23) => proper time elapsed on clocks at rest on the geoid is identical with the coordinate time t’’.  This is the correct way to express the fact that ideal clocks at rest on the geoid provide all of our standard reference clocks.

S4 Realisation of Coordinate Time

We now drop the t‘’ and just use t with the understanding that unit of time is referenced to TAI on the rotating geoid with synchronisation established in an underlying, local inertial, reference frame.  We therefore write (23), which is in the ECI frame, as,
			(ds^2)=-(1-((2 (U-(Subscript[Φ, 0])))/(c^2))) (c^2)(dt^2)+(1+((2 U)/(c^2))) ((dr^2)+(r^2)((dΩ)^2)).							(24)
(ds^2)=-(c^2)((dτ)^2)=>	((dτ)^2)=[1-((2 (U-(Subscript[Φ, 0])))/(c^2))-(1+((2 U)/(c^2)))((v^2)/(c^2))](dt^2),			where (v^2)=(((dr^2)+(r^2)((dΩ)^2))/(dt^2))					(25) & (26)
			=>dτ=[1-(( (U-(Subscript[Φ, 0])))/(c^2))-((v^2)/(2 (c^2)))]dt.				keeping orders up to (c^(-2))				(27)
Solving (27) we can relate the increment of coordinate time (TAI) to the proper time increment given by the slowly moving clock.					=>(∫_path)dt=(∫_path)[1+(( (U-(Subscript[Φ, 0])))/(c^2))+((v^2)/(2 (c^2)))]dτ.									(28)

S5 Relativistic Effects on Satellite Clocks

Keplerian orbits

For U≃((G M)/r) , we can describe the satellites with Keplarian orbits, for which we have the standard equations,
			r=((a (1-(e^2)))/(1+e Cos λ)),		(29)		where		Cos λ=((Cos u-e)/(1-e Cos u))			(30)
			 =a (1-e Cosu)	(31)					
where a and e are the semi-major axis and eccentricity respectively while u and λ are the eccentric and true anomalies respectively.  To calculate the eccentric anomaly, one must solve,
			u-e sin u=(Sqrt[((G M)/(a^3))]) (t-(Subscript[t, p])),									(32)
where (Subscript[t, p]) is the coordinate time of perigee passage.  If we apply conservation of energy and momentum at the apogee and and perigee, we also have,
			(E/m)=(((Subscript[v, a])^2)/2)-((G M)/(Subscript[r, a]))=(((Subscript[v, p])^2)/2)-((G M)/(Subscript[r, p])),			and 		(Subscript[v, p])(Subscript[r, p])=(Subscript[v, a]) (Subscript[r, a])
			=>(((Subscript[v, a])^2)/2)=G M((1/(Subscript[r, a]))-(1/(Subscript[r, p]))) (((1-(((Subscript[r, a])^2)/((Subscript[r, p])^2))))^(-1)),		where		(Subscript[r, a])=a (1+e),	(Subscript[r, p])=a (1-e).
				=G M ((-2 e a)/((a^2)(1-(e^2)))) [((((1-e))^2)/(-4e))],
				=G M(((1-e))/(2 a (1+e)))				=>(E/m)=G M [((1-e)/(2 a (1+e)))-(1/(a (1+e)))]=-G M (((1+e))/(2 a (1+e)))=-((G M)/(2 a)).
			=>((v^2)/2)-((G M)/r)=-((G M)/(2 a)).										(33)

Relativistic Effects

Quadrupole Contributions

- For atomic clocks, we use ECI => Sagnac effect is irrelevent (still required when dealing with ground-based receivers).
- Gravitational frequency shifts and second order Doppler shifts must be taken into account together.  
- In (28) (Subscript[Φ, 0]) includes the correction for using clocks at rest on the geoid (instead of at infinity). 
- From (18) we can see that the quadrupole contributes to (Subscript[Φ, 0]) in the term ((G M (Subscript[J, 2]))/(2 R))~3.76×(10^(-13))=> this must be accounted for in GPS. 
- From (13) we also see that in U, the quardupole component falling off particularly fast, (Subscript[U, quad])=((G M (R^2))/(r^3))(Subscript[J, 2]) (Subscript[P, 2])(Cosθ)~(10^(-14)) for GPS, which we can neglect => U≃((G M)/r)=> Keplarian orbits are applicable.

Clock rates for circular orbits

Recall (28) 		Δt=(∫_path)[1+(( (U-(Subscript[Φ, 0])))/(c^2))+((v^2)/(2 (c^2)))]dτ,								(28)
				=(∫_path)[1+((G M)/(r (c^2)))-(( (Subscript[Φ, 0]))/(c^2))+((G M)/(c^2)) ((1/r)-(1/(2 a)))]dτ,
				=(∫_path)[1-(( (Subscript[Φ, 0]))/(c^2))+((2G M)/(c^2)) ((1/r)-(1/(4 a)))]dτ,
				=(∫_path)[1+((3 G M)/(2 a (c^2)))-(( (Subscript[Φ, 0]))/(c^2))+((2 G M)/(c^2)) ((1/r)-(1/( a)))]dτ,						(34)
where if we put in numerical values for the first two components, we have
		((3 G M)/(2 a (c^2)))-(( (Subscript[Φ, 0]))/(c^2))=2.5046×(10^(-10))-6.9693×(10^(-10))=-4.4647×(10^(-10)).					(35a)
		=> for circular orbits the difference in ticks of a clock on the geoid and a clock in orbit is a factor of -4.4647×(10^(-10)), i.e.,
							t=(1+((3 G M)/(2 a (c^2)))-(( (Subscript[Φ, 0]))/(c^2)))τ+(Subscript[τ, 0]),					(35b)
		where (Subscript[τ, 0]) is a constant of integration set by synchronization of the orbiting clock.
		=> Time-dilation (second term in 35b) and gravitational blueshift (which dominates - third term in (35b))
Negative sign => standard clock in orbit (dτ in ECI) is beating two fast because it’s frequency is gravitationally blueshifted.
		=> for satellite clock to appear to an observer on the geoid to beat at the chosen frequency of 10.23MHz, the satellite clocks
		are adjusted lower in frequency so that the proper frequency is 
				[1-4.4647×(10^(-10))]×10.23 M Hz=10.22999999543 M Hz,
		this is done on the ground before the clock is placed in orbit.

Clock rates for eccentric orbits

For eccentric orbits (so when the last term in (34) does not vanish) we can use (32) to calculate,
			((d u)/(d t))=(Sqrt[((G M)/(a^3))])(((1-e Cos u))^(-1)),
			=>(∫_path)((2 G M)/(c^2)) ((1/r)-(1/( a)))dτ≃((2 G M)/(c^2)) (∫_path) ((1/r)-(1/( a)))dt,		as (28) =>dt=[1+O((c^(-2)))]dτ,
						=((2 G M)/(a (c^2))) (∫_path) (((e Cos u)/(1-e Cos u)))dt,		as (31) => r=a (1-e Cos u),
						=((2 G M)/(a (c^2))) (∫_path) (Sqrt[((a^3)/(G M))])((du/dt))e Cos u dt,
						=((2 (Sqrt[a G M]))/(c^2)) e (Sin u -Sin (Subscript[u, 0])),
						=((2 (Sqrt[a G M]))/(c^2)) e Sin u + const,						(38a)
			=>t=(1+((3 G M)/(2 a (c^2)))-(( (Subscript[Φ, 0]))/(c^2))+((2 (Sqrt[a G M]))/(c^2)) e Sin u)τ+(Subscript[τ, 1])						(38b)
As u = u(t), (38) gives us a relativistic correction for eccentric orbits that varys in time, which numerically gives,
			(Subscript[Δt, r])=4.4428×(10^(-10))e (Sqrt[a]) Sin u   ((s/(Sqrt[m]))).							(39)
- For a satellite of e = 0.001, this can reach up to 23 ns (taking a~2.5×(10^7)m) => must be accounted for.  
- Historically due to limited computational power, this correction is added by the receiver in GPS whereas GLONASS corrects for it before broadcast. 
- Looking at (38b), we can see a time dilation effect produced by orbital motion (second term), a gravitational blue(red)shift associated with the transfer of the time signal from high to low altitude (third term), and a time-dependent correcton associated with the orbital eccentricity (final term).