5.5 Theory of Pulsar Timing System
337
5.5.3 Evaluation of Pulsar Timing Stability
Generally, Allan variance is used to analyze the frequency stability of atomic clocks.
For the atomic clocks with linear frequency drift and low-frequency noise (also called
red noise), Hadamard variance is usually used to analyze the frequency stability. For
the pulsar timing, the methods used to analyze the stability of atomic clocks cannot
be directly used because of the irregularity of radio pulsar observation data interval.
For this reason, using the third-order polynomial fitting of pulsar’s timing residuals, a
statistical analysis method similar to the improved Hadamard variance was proposed
by Joseph H. Taylor in 1991, which is called Sigma-z variance (σ
2
z ). The Sigmaz variance is not sensitive to the randomness of pulsar observation and is an ideal
statistical method to compare the stability of the pulsar time with that of the atomic
time, mainly used to characterize the influence of the low-frequency pulsar timing
residuals on the stability of frequency. The so-called timing residuals refer to the
timing errors remained still after the phase, frequency and its rate, angular position
and proper motion in the pulsar timing model have been determined, also known as
timing noise.
By determining a set of polynomial coefficients p j (j = 0, 1, 2, 3), and utilizing the
frequency measurement residual sequence r(t i ) in the pulsar timing at observation
time t i (i = 1, 2, . . . , N ), the power spectral density of the pulsar timing residual can
be estimated. The parameters c j can be fitted by the method of least squares, that is
χ
2
=
N
i=1
⎡
⎣ r(t i ) −
3
j=0
c j p j (t i )
⎤
⎦ = min,
(5.89)
where c 0 , c 1 and c 2 are, respectively, the pulsar’s phase, period and its rate, and c 3 is
the estimating parameter of power spectral density of the residual, that is,
S 1 = (c 3 )
2
.
(5.90)
The corresponding frequency is f ≈
1
T
, where T = t N − t 1 , representing the
observational time span.
The frequency measurement residual sequence r(t i ) with the observational time
span of T is divided into m subsequences with approximately equal length, and then
the every subsequence is divided once again using the same way. As a result, m new
parameter-estimating values c 3 will be obtained, and thus the power spectral density
is
S m =
1
m
m
k=1
(c 3 )
2
k .
(5.91)
The corresponding frequency is f ≈
m
T
, where m = 2, 4, 8,….
337
5.5.3 Evaluation of Pulsar Timing Stability
Generally, Allan variance is used to analyze the frequency stability of atomic clocks.
For the atomic clocks with linear frequency drift and low-frequency noise (also called
red noise), Hadamard variance is usually used to analyze the frequency stability. For
the pulsar timing, the methods used to analyze the stability of atomic clocks cannot
be directly used because of the irregularity of radio pulsar observation data interval.
For this reason, using the third-order polynomial fitting of pulsar’s timing residuals, a
statistical analysis method similar to the improved Hadamard variance was proposed
by Joseph H. Taylor in 1991, which is called Sigma-z variance (σ
2
z ). The Sigmaz variance is not sensitive to the randomness of pulsar observation and is an ideal
statistical method to compare the stability of the pulsar time with that of the atomic
time, mainly used to characterize the influence of the low-frequency pulsar timing
residuals on the stability of frequency. The so-called timing residuals refer to the
timing errors remained still after the phase, frequency and its rate, angular position
and proper motion in the pulsar timing model have been determined, also known as
timing noise.
By determining a set of polynomial coefficients p j (j = 0, 1, 2, 3), and utilizing the
frequency measurement residual sequence r(t i ) in the pulsar timing at observation
time t i (i = 1, 2, . . . , N ), the power spectral density of the pulsar timing residual can
be estimated. The parameters c j can be fitted by the method of least squares, that is
χ
2
=
N
i=1
⎡
⎣ r(t i ) −
3
j=0
c j p j (t i )
⎤
⎦ = min,
(5.89)
where c 0 , c 1 and c 2 are, respectively, the pulsar’s phase, period and its rate, and c 3 is
the estimating parameter of power spectral density of the residual, that is,
S 1 = (c 3 )
2
.
(5.90)
The corresponding frequency is f ≈
1
T
, where T = t N − t 1 , representing the
observational time span.
The frequency measurement residual sequence r(t i ) with the observational time
span of T is divided into m subsequences with approximately equal length, and then
the every subsequence is divided once again using the same way. As a result, m new
parameter-estimating values c 3 will be obtained, and thus the power spectral density
is
S m =
1
m
m
k=1
(c 3 )
2
k .
(5.91)
The corresponding frequency is f ≈
m
T
, where m = 2, 4, 8,….