5.1 Principle of Pulsar Navigation
275
denote the right ascension and declination of the pulsar PSR in the SSB coordinate system, also called pulsar’s angular position; R SC , R E and R SC/E , in the SSB
coordinate system, respectively, denote the position vectors of the spacecraft, Earth
and spacecraft relative to the Earth; b denotes the position vector of the SSB in the
Sun-mass-centric coordinate system O S −X S Y S Z S ; Q denotes the projection of the
spacecraft’s position along the direction from the SSB to the pulsar; n denotes the
position unit vector of the pulsar, also called angular position vector, namely,
n =
cos α cos λ cos α sin λ sin α
T
(5.1)
Because pulsars are extremely far away from the solar system, reaching thousands
of light-years or even tens of thousands of light-years, the n can generally be regarded
as a constant vector for the whole solar system. In the SSB coordinate system, the
right ascensions and declinations of the pulsars can also be measured accurately [1].
In other words, the direction vectors of the pulsars in the SSB coordinate system are
known.
For a spacecraft equipped with X-ray detection device and atomic clock, as a pulsar
rotates and its magnetic pole beam sweep over the detection device, the spacecraft
will receive an X-ray pulse signal and the arrival-time of pulse signal, t SC , is recorded
with the atomic clock. Meanwhile, the time of the same pulse signal arriving at the
SSB, t SSB , can be accurately predicted by pulsar timing model. For the whole solar
system, the beams emitted from pulsars can be regarded as plane waves. Thereby, in
Fig. 5.1, the time when the same pulse signal arrives at spacecraft SC is same as that
when it arrives at point Q. According to the trigonometric projection relation, and
considering a bias of the onboard clock, the measurement equation of the XPNAV
can be expressed as
c(t SSB − t SC ) = n · R SC + c · δt ϕ ,
(5.2)
where R SC =
x y z
T , c is the light velocity, and δt ϕ is the bias of the onboard
clock.
There are four unknowns in Eq. (5.2), including three-position coordinate components of the spacecraft and one clock bias. When four pulsars are observed at the
same time, then four measurement equations can be established to solve the four
unknowns. It is referred to as geometric orbit determination. Combining with the
model of orbital mechanics of the spacecraft, it is only needed to observe one pulsar
during each observation arc-segment, and then the position and time parameters of the
spacecraft can be determined, which is referred to as dynamics orbit determination.
Obviously, the principle of the XPNAV is very similar to that of the GNSS, and
their basic observables are distances. The distance measurement of the XPNAV is
determined by the time differences between the same X-ray pulse signal arriving,
respectively, at the SSB and spacecrafts, while that of the GNSS is determined by
the arrival-time from the pseudo-random noise signals broadcasted by the navigation
satellites to the user terminals. At present, the navigation accuracy using the GNSS is
275
denote the right ascension and declination of the pulsar PSR in the SSB coordinate system, also called pulsar’s angular position; R SC , R E and R SC/E , in the SSB
coordinate system, respectively, denote the position vectors of the spacecraft, Earth
and spacecraft relative to the Earth; b denotes the position vector of the SSB in the
Sun-mass-centric coordinate system O S −X S Y S Z S ; Q denotes the projection of the
spacecraft’s position along the direction from the SSB to the pulsar; n denotes the
position unit vector of the pulsar, also called angular position vector, namely,
n =
cos α cos λ cos α sin λ sin α
T
(5.1)
Because pulsars are extremely far away from the solar system, reaching thousands
of light-years or even tens of thousands of light-years, the n can generally be regarded
as a constant vector for the whole solar system. In the SSB coordinate system, the
right ascensions and declinations of the pulsars can also be measured accurately [1].
In other words, the direction vectors of the pulsars in the SSB coordinate system are
known.
For a spacecraft equipped with X-ray detection device and atomic clock, as a pulsar
rotates and its magnetic pole beam sweep over the detection device, the spacecraft
will receive an X-ray pulse signal and the arrival-time of pulse signal, t SC , is recorded
with the atomic clock. Meanwhile, the time of the same pulse signal arriving at the
SSB, t SSB , can be accurately predicted by pulsar timing model. For the whole solar
system, the beams emitted from pulsars can be regarded as plane waves. Thereby, in
Fig. 5.1, the time when the same pulse signal arrives at spacecraft SC is same as that
when it arrives at point Q. According to the trigonometric projection relation, and
considering a bias of the onboard clock, the measurement equation of the XPNAV
can be expressed as
c(t SSB − t SC ) = n · R SC + c · δt ϕ ,
(5.2)
where R SC =
x y z
T , c is the light velocity, and δt ϕ is the bias of the onboard
clock.
There are four unknowns in Eq. (5.2), including three-position coordinate components of the spacecraft and one clock bias. When four pulsars are observed at the
same time, then four measurement equations can be established to solve the four
unknowns. It is referred to as geometric orbit determination. Combining with the
model of orbital mechanics of the spacecraft, it is only needed to observe one pulsar
during each observation arc-segment, and then the position and time parameters of the
spacecraft can be determined, which is referred to as dynamics orbit determination.
Obviously, the principle of the XPNAV is very similar to that of the GNSS, and
their basic observables are distances. The distance measurement of the XPNAV is
determined by the time differences between the same X-ray pulse signal arriving,
respectively, at the SSB and spacecrafts, while that of the GNSS is determined by
the arrival-time from the pseudo-random noise signals broadcasted by the navigation
satellites to the user terminals. At present, the navigation accuracy using the GNSS is