5.6 Methods of Large-Scale Navigation
357
5.6.5 Determination of Orbit and Time Parameters
The orbit and time parameters of spacecrafts can be determined by using the X-ray
signals radiated from pulsars. According to the different basic observables and navigation ways, there are four types of measurement equations, including the equation
of directly using distance, that of using pulse phase, that for absolute navigation, and
that for relative navigation.
5.6.5.1 Measurement Equation of Directly Using Distance
Since the rotation periods of pulsars are extremely stable, the X-ray photons continuously radiated from the magnetic poles of pulsars can be gathered into the pulse
signals as arriving at the spacecrafts. For a certain pulse signal, would the time when
the pulsar emits the signals be known, the measurement equation can be established
directly using the ranging observables, that is
ρ = n SC · (D − p) + d Rel ,
(5.109)
where ρ = c(t SC − t T ); c is the velocity of light; t SC is the time when the X-ray pulse
signal emitted from the pulsar arrives at the spacecraft under the timescale TCB; t T is
the time when the pulsar emits the X-ray pulse signal under the timescale TCB; n SC
is the direction vector from the spacecraft to the pulsar; p and D are, respectively,
the position vectors of the spacecraft and pulsar in the BCRS; d Rel is the correction
of relativistic effect.
If the position error of pulsar, the time bias of pulsar emitting pulse signal, the
residual of relativistic effect and other errors are collectively called measurement
noise, and the spacecraft’s clock bias and its position correction are considered, then
there is the following formula.
δρ d = −n SC · δ p − cδt SC + η d ,
(5.110)
where δρ d = c
˜ t SC − t T
− n SC · (D − ˜
p) − d Rel ; ˜ t SC and δt SC are, respectively, the
onboard clock reading time and its bias under the timescale TCB; ˜
p and δp are, respectively, the approximate position vector and its correction; η d is the measurement
noise.
Obviously, the basic observable of Eq. (5.110) is the modified distance δρ d from
the signal emitting source (the pulsar) to the user’s position (the spacecraft), which
is consistent with the GNSS measurement equation using the pseudorange. When
more than four pulsars are observed at the same time, the position correction and
clock bias parameters of the spacecraft can be directly estimated by using the method
of least squares. Although the time when the pulse signals arrive at the spacecrafts
can be measured, the pulsars belong to natural celestial bodies, and the time when
the pulsars emit the pulse signals is unable to be known exactly. That is to say, the
357
5.6.5 Determination of Orbit and Time Parameters
The orbit and time parameters of spacecrafts can be determined by using the X-ray
signals radiated from pulsars. According to the different basic observables and navigation ways, there are four types of measurement equations, including the equation
of directly using distance, that of using pulse phase, that for absolute navigation, and
that for relative navigation.
5.6.5.1 Measurement Equation of Directly Using Distance
Since the rotation periods of pulsars are extremely stable, the X-ray photons continuously radiated from the magnetic poles of pulsars can be gathered into the pulse
signals as arriving at the spacecrafts. For a certain pulse signal, would the time when
the pulsar emits the signals be known, the measurement equation can be established
directly using the ranging observables, that is
ρ = n SC · (D − p) + d Rel ,
(5.109)
where ρ = c(t SC − t T ); c is the velocity of light; t SC is the time when the X-ray pulse
signal emitted from the pulsar arrives at the spacecraft under the timescale TCB; t T is
the time when the pulsar emits the X-ray pulse signal under the timescale TCB; n SC
is the direction vector from the spacecraft to the pulsar; p and D are, respectively,
the position vectors of the spacecraft and pulsar in the BCRS; d Rel is the correction
of relativistic effect.
If the position error of pulsar, the time bias of pulsar emitting pulse signal, the
residual of relativistic effect and other errors are collectively called measurement
noise, and the spacecraft’s clock bias and its position correction are considered, then
there is the following formula.
δρ d = −n SC · δ p − cδt SC + η d ,
(5.110)
where δρ d = c
˜ t SC − t T
− n SC · (D − ˜
p) − d Rel ; ˜ t SC and δt SC are, respectively, the
onboard clock reading time and its bias under the timescale TCB; ˜
p and δp are, respectively, the approximate position vector and its correction; η d is the measurement
noise.
Obviously, the basic observable of Eq. (5.110) is the modified distance δρ d from
the signal emitting source (the pulsar) to the user’s position (the spacecraft), which
is consistent with the GNSS measurement equation using the pseudorange. When
more than four pulsars are observed at the same time, the position correction and
clock bias parameters of the spacecraft can be directly estimated by using the method
of least squares. Although the time when the pulse signals arrive at the spacecrafts
can be measured, the pulsars belong to natural celestial bodies, and the time when
the pulsars emit the pulse signals is unable to be known exactly. That is to say, the