Flelativistic effects

106 6. Observables Table 6.3. Correction term 6R in meters for refined Saastamoinen model

Zenith Station height above sea level [km]

distance 0 0.5 1.0 1.5 2.0 3.0 4.0 5.0

60°00' 0.003 0.003 0.002 0.002 0.002 0.002 0.001 0.001 66°00' 0.006 0.006 0.005 0.005 0.004 0.003 0.003 0.002 70°00' 0.012 0.011 0.010 0.009 0.008 0.006 0.005 0.004 73°00' 0.020 0.018 0.017 0.015 0.013 0.011 0.009 0.007 75°00' 0.031 0.028 0.025 0.023 0.021 0.017 0.014 0.011 76°00' 0.039 0.035 0.032 0.029 0.026 0.021 0.017 0.014 77°00' 0.050 0.045 0.041 0.037 0.033 0.027 0.022 0.018 78°00' 0.065 0.059 0.054 0.049 0.044 0.036 0.030 0.024 78°30' 0.075 0.068 0.062 0.056 0.051 0.042 0.034 0.028 79°00' 0.087 0.079 0.072 0.065 0.059 0.049 0.040 0.033 79°30' 0.102 0.093 0.085 0.077 0.070 0.058 0.047 0.039 79°45' 0.111 0.101 0.092 0.083 0.076 0.063 0.052 0.043 80°00' 0.121 0.110 0.100 0.091 0.083 0.068 0.056 0.047

angles, see Lanyi (1984)', since the tropospheric path delay at zenith is am-plified by the mapping function.

The difficulty in modeling the tropospheric effect will require continu-ation of research and development for some years. Due to the opinion of Lanyi (1984), the best solution is to combine surface and radio sonde mete-orological data, water vapor radiometer measurements and statistics. This is a major task and an appropriate model has not yet been found.

6.4 Itelativistic effects transformation

x - vt x' = ---r===::=

)1-~

y'

=

^y

Zl = Z

t- ~x

t'

=

^c

)1-~

X,

+

v· t'

X=

)1-~

y

=

^y'

z = Zl

t'

+

^{V x'}

t= CT

)1-~

107

(6.116)

where c is the speed of light. An elegant and simple derivation of these formulas can be found in Joos (1956), pp. 217-218. By using Eqs. (6.116), the relation

(6.117) may be verified. This means that the norm of a vector in space-time coor-dinates is invariant with respect to the choice of its reference system. Note that in case of c = ⁰⁰ the Lorentz transformation converts to the Galilei transformation

x' = x - vt y'

=

^y

(6.118)

Zl = z t'

=

^t

which is fundamental in classical mechanics.

The theory of special relativity is by definition restricted to inertial sys-tems. The application of the Lorentz transformation reveals some features of that theory.

Time dilation. Consider two time events tl and t2 at the same location x of the system at rest. Due to the Lorentz transformation (6.116), the corresponding events in the moving system take the form

I tl -

?:

^{x I t2 -}

?:

^X

tl = t2 = (6 119)

)1- ^~ )1- ^~ .

Denoting the time interval in the moving system by t1t' = t~ - t~ and in the resting system by t1t = t2 - t1, the difference of the two expressions in (6.119) yields the time dilation

(6.120)

108 6. Observables which means that the time interval dt in S for an observer moving with S' is lengthened to dt'. IT the time intervals are monitored by clocks, the moving clocks run slower than resting clocks. For the inverse situation the same holds so that an intervall dt' in S' is lengthened to dt for a resting observer.

Lorentz contraction. The derivation of the Lorentz contraction is analogous to the time dilation. Considering now two locations Xl and X2 in the resting system S at the same epoch t, then the corresponding locations in the movig system S' are given by the Lorentz transformation. Using the abbreviations dx

=

^{X2 - Xl and dX'}

=

^{x~ - x~,} then it infers from above that dX' is lengthened to dx for a resting observer. Expressing it in another way, for a resting observer the dimension of a moving body seems to be contracted:

dX'

=

^dx

V ~

^{1 -}

C2 .

^(6.121)

Second-orner Doppler effect. Since frequency is inversely proportional to time, one can deduce immediately from the considerations on time dilation that the frequency

I'

of a moving emitter would be reduced to

I

when received by an resting observer. This is the second-order Doppler effect given by the formula

I , I

-10'

^{1 -(!I v2 (6.122)}

Mass relation. Special relativity also affects masses. Denoting the masses in the two reference frames Sand S' by m and

m',

^then

(6.123)

is the corresponding mass relation, cf. for example Heckmann (1985).

Each of the formulas (6.120) to (6.123) comprises the same square root which may be expanded into binomial series:

1 1

(V)

²

J1-~ =

1+

2"

~

(6.124) /1- ::

=

^1-

~ (~)

²

6.4 Relativistic effects 109 Substituting these expansions into Eqs. (6.120) to (6.123), each related to an observer at rest, then

l:l.t' - l:l.t = l:l.x' - l:l.x = _

I' - I

= m' - m = _ ~ (~) ²

l:l.t l:l.x

I

m 2 c (6.125)

accounts for the mentioned effects of the special relativity in one formula.

6.4.2 General relativity

The theory of general relativity includes accelerated reference systems too, where the gravitational field plays the key role. Formulas analogous to (6.125) may be derived when one replaces the kinetic energy! v² in spe-cial relativity by the potential energy l:l.U. Thus,

l:l. t' - l:l. t l:l.t

l:l.x' - l:l.x

l:l.x _{- - -}

I' - I I

m'-m

m (6.126)

represents the relations in general relativity, cf. Heckmann (1985), where l:l.U is the difference of the gravitational potential in the two reference frames under consideration.

6.4.3 Relevant relativistic effects for GPS

The reference frame (relatively) at rest is located in the center of the earth and an accelerated reference frame is attached to each GPS satellite. There-fore, the theory of special and general relativity must be taken into account.

Relativistic effects are relevant for the satellite orbit, the satellite signal propagation, and both the satellite and receiver clock. An overview of all these effects is given for example in Zhu and Groten (1988), the relativistic effects on rotating and gravitating clocks is also treated in Grafarend and Schwarze (1991). With respect to general relativity, Ashby (1987) shows that only the gravitational field of the earth must be considered. Sun and moon and consequently all other masses in the solar system are negligible.

Relativity affecting the satellite orbit. The gravitational field of the earth causes also relativistic perturbations in the satellite orbits. An approximate formula for the disturbing acceleration is given by Eq. (4.46). For more de-tails the reader is referred to Zhu and Groten (1988).

Relativity affecting the satellite signal. The gravitational field gives rise to a space-time curvature of the satellite signal. Therefore, a propagation

cor-110 6. Observables rection must be applied to get the Euclidean range for instance. The range correction modeled by Holdridge (1967) takes the form

brei

=

^{2J.l In}

rI + ei + uf.

c²

ei ⁺ ei - ei

^(6.127)

where J.l is the earth's. gravitational constant. The geoc~ntric distances of satellite j and observing site i are denoted

rI

^and

ei,

^and

ei

is the distance be-tween the satellite and the observing site. In order to estimate the maximum effect for a point on the earth's surface take the mean radius RE = 6370km and a mean altitude of h = 20200 km for the satellites. The maximum distance

ei

results from the Pythagorean theorem and is about 25800 km.

Substituting these values, the maximum range error brel = 18.7 mm results from (6.127). Note that this maximum value only applies to point posi-tioning. In relative positioning the effect is much smaller and amounts to 0.001 ppm, d. Zhu and Groten (1988).

Relativity affecting the satellite clock. The fundamental frequency fo of the satellite clock is 10.23 MHz. All the signals are based on this frequency which is influenced by the motion of the satellite and by the difference of the gravitational field at the satellite and the observing site. The correspond-ing effects of special and general relativity are small and may be linearly superposed. Thus,

brel

==

fti - fo

=! (!!.)2 ^{+ ~U}

fo 2 ^C c²

special

+

general (6.128)

relativity

is the effect on the frequency of the satellite clock where Eqs. (6.125) and (6.126) have been used. To get a numerical value, circular orbits and a spherical earth with the observing site on its surface are assumed. Backing on these simplifications, (6.128) takes the form

brel

==

fb - fo =

! (!!.) 2 ⁺ ~ [

^{1 __ 1_] (6.129)}

fo 2 c c² RE

+

^{h RE}

with v being the satellite's mean velocity. Substituting numerical values yields

fti

!o

^fo

=

^4.464.10-10 (6.130)

which, despite the simplifications, is sufficiently accurate. Ashby (1987) for instance takes into account the J2-term for the potential and the centrifugal

6.5 Multipath 111 forces and gets the only slightly different result 4.465.10-10 . Recall that

fb

is the emitted frequency and fo is the frequency received at the observation site. Thus, it can be seen that the satellite transmitted nominal frequency would be increased by df = 4.464.10-10 • fo = 4.55.10-³ Hz. However, it is desired to receive the nominal frequency. This is achieved by an offset df in the satellite clock frequency, so that 10.22999999545 MHz are emitted, cf. Spilker (1980).

Another small periodic effect arises due to the assumption of a circu-lar orbit (which is almost true for the Block II satellites). An adequate correction formula is given by Gibson (1983) as

6^rel =

~..Jiiii

_c ^{e sin E (6.131)}

where e denotes the eccentricity, a the semimajor axis, and E the eccentric anomaly. This relativistic effect is taken into account by the clock polyno-mial broadcast via the navigation message, cf. Sect. 4.4.2, where the time dependent eccentric anomaly E is expanded into a Taylor series. Thus, Eq. (6.131) gives slightly more accurate results. However, the decision to use this formula must be correlated with a correction of the clock poly-nomial coefficients with respect to the built in accounting for the periodic effect. Van Dierendonck et al. (1980) present the necessary formulas. In case of relative positioning, the effect cancels out, cf. Zhu and Groten (1988).

Relativity affecting the receiver clock. A receiver clock located at the earth's surface is rotating with respect to the resting reference frame at the geocen-ter. The associated linear velocity at the equator is approximately 0.5 km·s-¹ and thus roughly one tenth of the satellite's velocity. Substituting this value into the special relativistic part of Eq. (6.128) yields a relative frequency shift in the order of 10-¹² which after 3 hours corresponds to a clock error of 10 nanoseconds (1 ns = 10-⁹ s == 30 cm). Ashby (1987) presented the correc-tion formula for this Sagnac effect; however, since usually the correccorrec-tion is performed by the receiver software the formula is not explicitly given here.

Trong tài liệu B. Hofmann-Wellenhof, H. Lichtenegger, and 1. Collins (Trang 121-126)