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5 X-ray Pulsar-Based Navigation: Theories and Experiments
According to the definition of TAI and TT, the TAI takes the length of second of
the SI units on the rotating geoid as the unit, which is regarded as the coordinate
timescale in the ECEF reference system, and is also a concrete realization form of
the ideal timescale TT. That is to say, in the GCRS, the proper time, metered along
the world-line on the geoid and relative to the at-rest observer and its carried clock,
is defined as the TAI. The basic relation between the idea TT and TAI is expressed
as
TT = TAI + 32
s
.184.
(5.99)
However, for the atomic clocks carried by the spacecrafts, there are orbital velocity
relative to the GCRS and gravitational potential relative to the geoid, and thus,
TT =
1 −
1
c 2
W G −
3μ E
2a
(τ − τ 0 ) +
2
c 2
√ μ E a(sin E − sin E 0 ),
(5.100)
where τ is the proper time in the spacecraft-body coordinate system, which has
been transformed to the TAI; τ 0 is the clock reading as the TT is equal to zero; c
is the velocity of light in vacuum; W G is the gravitational potential on the geoid,
which is the standardized definition constant of IERS 2000, with the value of
62636856.0 m
2 /s
2 ; μ E is the gravitational constant of the Earth; a is the semi-major
axis of the spacecraft’s elliptic orbit; E is the eccentric anomaly of the spacecraft’s
orbit.
The first term on the right side of formula (5.100) is a long-term one, and the
second term is a periodic one. In order to avoid the long-term drift of the onboard
clock relative to the TT, the clock’s frequency must be adjusted, which is the effect
from the long-term term of formula (5.100). Supposed that the reference frequency
of the onboard clock is f 0 , the frequency before launching the spacecraft should be
adjusted to
f = (1 − k)f 0 ,
(5.101)
where k =
1
c 2
W G −
3μ E
2a
.
For example, the semi-major axis is 26570 km for the GPS satellite orbits, the reference frequencies of their onboard clocks are 10.23 MHz, and thus k = 4.472×10
−10 .
In other words, before launching GPS satellites, the clock reference frequencies are
adjusted to 10.22999999543 MHz, to eliminate the long-term effect of the relativity.
(4) Transformation from TT to TCG
According to the resolution of IAU general assembly in 1991, the transformation
from the TT to TCG is linear, and the difference between them is a proportional
constant, i.e.,
TCG = TT + L G × (JD − 2443144.5) × 86400,
(5.102)
5 X-ray Pulsar-Based Navigation: Theories and Experiments
According to the definition of TAI and TT, the TAI takes the length of second of
the SI units on the rotating geoid as the unit, which is regarded as the coordinate
timescale in the ECEF reference system, and is also a concrete realization form of
the ideal timescale TT. That is to say, in the GCRS, the proper time, metered along
the world-line on the geoid and relative to the at-rest observer and its carried clock,
is defined as the TAI. The basic relation between the idea TT and TAI is expressed
as
TT = TAI + 32
s
.184.
(5.99)
However, for the atomic clocks carried by the spacecrafts, there are orbital velocity
relative to the GCRS and gravitational potential relative to the geoid, and thus,
TT =
1 −
1
c 2
W G −
3μ E
2a
(τ − τ 0 ) +
2
c 2
√ μ E a(sin E − sin E 0 ),
(5.100)
where τ is the proper time in the spacecraft-body coordinate system, which has
been transformed to the TAI; τ 0 is the clock reading as the TT is equal to zero; c
is the velocity of light in vacuum; W G is the gravitational potential on the geoid,
which is the standardized definition constant of IERS 2000, with the value of
62636856.0 m
2 /s
2 ; μ E is the gravitational constant of the Earth; a is the semi-major
axis of the spacecraft’s elliptic orbit; E is the eccentric anomaly of the spacecraft’s
orbit.
The first term on the right side of formula (5.100) is a long-term one, and the
second term is a periodic one. In order to avoid the long-term drift of the onboard
clock relative to the TT, the clock’s frequency must be adjusted, which is the effect
from the long-term term of formula (5.100). Supposed that the reference frequency
of the onboard clock is f 0 , the frequency before launching the spacecraft should be
adjusted to
f = (1 − k)f 0 ,
(5.101)
where k =
1
c 2
W G −
3μ E
2a
.
For example, the semi-major axis is 26570 km for the GPS satellite orbits, the reference frequencies of their onboard clocks are 10.23 MHz, and thus k = 4.472×10
−10 .
In other words, before launching GPS satellites, the clock reference frequencies are
adjusted to 10.22999999543 MHz, to eliminate the long-term effect of the relativity.
(4) Transformation from TT to TCG
According to the resolution of IAU general assembly in 1991, the transformation
from the TT to TCG is linear, and the difference between them is a proportional
constant, i.e.,
TCG = TT + L G × (JD − 2443144.5) × 86400,
(5.102)