# In-orbit demonstration of X-ray pulsar navigation with the _Insight_ -HXMT
satellite

S. J. Zheng1, S. N. Zhang1,2, F. J. Lu1,2, W. B. Wang3, Y. Gao3, T. P.
Li1,2,4, L. M. Song1,2, M. Y. Ge1, D. W. Han1, Y. Chen1, Y. P. Xu1, X. L.
Cao1, C. Z. Liu1, S. Zhang1,2, J. L. Qu1,2, Z. Chang1, G. Chen1, L. Chen5, T.
X. Chen1, Y. B. Chen4, Y. P. Chen1, W. Cui1,4, W. W. Cui1, J. K. Deng4, Y. W.
Dong1, Y. Y. Du1, M. X. Fu4, G. H. Gao1,2, H. Gao1,2, M. Gao1, Y. D. Gu1, J.
Guan1, C. Gungor1, C. C. Guo1,2, D. W. Han1, W. Hu1, Y. Huang1, J. Huo1, J. F.
Ji4, S. M. Jia1,2, L. H. Jiang1, W. C. Jiang1, J. Jin1, Y. J. Jin6, B. Li1, C.
K. Li1, G. Li1, M. S. Li1, W. Li1, X. Li1, X. B. Li1, X. F. Li1, Y. G. Li1, Z.
J. Li1,2, Z. W. Li1, X. H. Liang1, J. Y. Liao1, G. Q. Liu4, H. W. Liu1, S. Z.
Liu1, X. J. Liu1, Y. Liu1, Y. N. Liu6, B. Lu1, X. F. Lu1, T. Luo1, X. Ma1, B.
Meng1, Y. Nang1,2, J. Y. Nie1, G Ou1, N. Sai1,2, R. C. Shang4, L. Sun1, Y.
Tan1, L. Tao1, W. Tao1, Y. L. Tuo1,2, G. F. Wang1, J. Wang1, W. S. Wang1, Y.
S. Wang1, X. Y. Wen1, B. B. Wu1, M. Wu1, G. C. Xiao1,2, S. L. Xiong1, H. Xu1,
L. L. Yan1,2, J. W. Yang1, S.Yang1, Y. J. Yang1, A. M. Zhang1, C. L. Zhang1,
C. M. Zhang1, F. Zhang1, H. M. Zhang1, J. Zhang1, Q. Zhang1, T. Zhang1, W.
Zhang1,2, W. C. Zhang1, W. Z. Zhang5, Y. Zhang1, Y. Zhang1,2, Y. F. Zhang1, Y.
J. Zhang1, Z. Zhang4, Z. Zhang6, Z. L. Zhang1, H. S. Zhao1, J. L. Zhao1, X. F.
Zhao1,2, Y. Zhu1, Y. X. Zhu1, C. L. Zou1 1 Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China<EMAIL_ADDRESS><EMAIL_ADDRESS>2 University of
Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049,
China  3 Technology and Engineering Center for Space Utilization Chinese
Academy of Sciences, Beijing 10094, China  4 Department of Physics, Tsinghua
University, Beijing 100084, China  5 Department of Astronomy, Beijing Normal
University, Beijing 100088, China  6 Department of Engineering Physics,
Tsinghua University, Beijing 100084, China

###### Abstract

In this work, we report the in-orbit demonstration of X-ray pulsar navigation
with _Insight_ -Hard X-ray Modulation Telescope (_Insight_ -HXMT), which was
launched on Jun. 15th, 2017. The new pulsar navigation method ‘Significance
Enhancement of Pulse-profile with Orbit-dynamics’ (SEPO) is adopted to
determine the orbit with observations of only one pulsar. In this test, the
Crab pulsar is chosen and observed by _Insight_ -HXMT from Aug. 31th to Sept.
5th in 2017. Using the 5-day-long observation data, the orbit of _Insight_
-HXMT is determined successfully with the three telescopes onboard $-$ High
Energy X-ray Telescope (HE), Medium Energy X-ray Telescope (ME) and Low Energy
X-ray Telescope (LE) $-$ respectively. Combining all the data, the position
and velocity of the _Insight_ -HXMT are pinpointed to within 10 km (3$\sigma$)
and 10 m s-1 (3$\sigma$), respectively.

stars: pulsar; techniques: miscellaneous

##  1 Introduction

Space navigation plays an important role for spacecraft launched to the earth
orbit, lunar and Mars vicinities, and deep space. At present, the navigation
of the spacecraft in low earth orbits mainly relies on global navigation
satellite systems (GNSS) and the ground-based tracking systems, while the
navigation in deep-space is mainly based on radio technologies (e.g. the US
deep-space network). With the increasing number of space tasks, the ground
stations are becoming overloaded and the operating costs are getting higher.
Navigation also can not be handled timely, due to the communication delay or
unexpected malfunction of ground stations. Pulsar navigation, an autonomous
navigation technology, is receiving more and more attention as it is less
dependent on the support of ground equipments and meets the continuous
navigation requirements for space missions in different orbits.

The first pulsar was discovered in 1967 (Hewish et al., 1968), and up until
today more than 2000 pulsars have been discovered with frequency covering from
radio, infrared, optical, ultraviolet, X-ray to gamma-ray. They are called
‘celestial GPS satellites’ because of their long-term timing stability which
is comparable to atomic clocks on the earth (Taylor, 1991). The pulsar
navigation using the radio and X-ray pulsars was proposed in 1974 (Downs,
1974) and in 1981 (Chester & Butman, 1981), respectively. A large number of
researches on the theory, algorithms and simulations have been carried out for
spacecrafts in earth orbits, Mars orbits and deep space orbits (e.g. Sheikh et
al (2007); Hanson et al (2008); Emadzadeh & Speyer (2011); Becker et al.
(2013); Wei et al. (2013); Wang et al. (2014); Shemar et al. (2016); Cui et
al. (2016)), and the tracking phase or positioning errors for different
pulsars are predicted in detail (Sheikh et al, 2007; Hanson et al, 2008). In
1999, the first in-orbit test was carried out with the Unconventional Stellar
Aspect (USA) on the ARGOS satellite (Wood, 1993; Sheikh, 2005). In 2016, the
results of testing pulsar navigation with Gamma-Ray Bursts Polarimeter
(POLAR), which was launched onboard the Chinese space laboratory TG-2 on Sept.
15th, 2016, were reported (Zheng et al., 2017). With the effective area of
$\sim$ 200 cm2 and wide field of view of more than 2$\pi$ Sr (Produit et al.,
2005; Li et al., 2018), POLAR has monitored the Crab pulsar for a long time.
With 31-day-long observations of the Crab pulsar, the TG-2 orbit was
determined successfully. The orbital elements were determined with the orbit
deviation within 20 km (1 $\sigma$). On Jun. 3th, 2017, the Neutron Star
Interior Composition Explorer (NICER) was launched on the International Space
Station (ISS). It comprises 56 identical X-ray telescopes with effective area
of 2000 cm2 in total, each of the telescopes consisting of a concentrating
X-ray optic and a single-pixel silicon drift detector (SDD) (Paul et al.,
2017). By measuring the Time of Arrival (TOAs) of five millisecond pulsars,
NICER pinpointed its location within 5 km (Alexandra, 2017).

On Jun. 15th, 2017, _Insight_ -Hard X-ray Modulation Telescope (_Insight_
-HXMT) was launched in China (Zhang et al., 2018; Li et al., 2018). It
consists of three X-ray slat-collimated telescopes: High Energy X-ray
Telescope (HE), Medium Energy X-ray Telescope (ME) and Low Energy X-ray
Telescope (LE). HE contains 18 cylindrical NaI(Tl)/CsI(Na) Phoswich detectors
with the energy band 20-250 keV and a total geometrical area of about 5000
cm2. ME is composed of 1728 Si-PIN detectors with the energy band 5-30 keV and
a total geometrical area of about 952 cm2. LE is composed of 96 Swept Charge
Device (SCD) detectors which are sensitive in the 1-15 keV with a total
geometrical area of about 384 cm2. Due to the wide energy band and large
detection area, _Insight_ -HXMT can also be used to make in-orbit test of
pulsar navigation. In this paper, the new pulsar navigation method
‘Significance Enhancement of Pulse-profile with Orbit-dynamics’ (SEPO), which
has been tested in POLAR (Zheng et al., 2017), is adopted to determine the
orbit of _Insight_ -HXMT autonomously. The Kalman Filter method can also be
used to make in-orbit demonstration of X-ray pulsar navigation (Wang et al.,
2018).

##  2 Navigation method

Conventionally, at least three pulsars are needed in pulsar navigation models
to estimate the position of a spacecraft. However, it results in complexity of
the payload and poses risks for control system. The one-pulsar navigation
method SEPO has been proposed by combining the significance analysis of pulse
profile and orbit dynamics. The orbital dynamics model can produce a high-
precision orbit forecast in a short period of time but with a long-term drift.
The pulse profile will be ‘deformed’ due to the orbit deviation as explained
below, resulting in the decrease of the significance of the profile signal,
which is used to update the orbital parameters continuously.

###  2.1 The orbit dynamics

The orbit is described by six orbital elements: semi-major axis ($a$),
eccentricity ($e$), inclination angle ($i$), right ascension of the ascending
node (RAAN, $\omega$), argument of perigee ($\Omega$), and mean anomaly ($w$).
Considering the altitude of the _Insight_ -HXMT orbit (about 550 km) and the
complexity of the earth’s perturbation force, the orbit forecast model is
constructed with the 70th order gravity field, the atmospheric drag, and the
solar radiation pressure, and the track integrator is Runge-Kutta-Fehlberg
7(8) with an integration step of 60 seconds (Wang & Gao, 2016). In such
conditions, the position error is less than 3.5 km for the forecast orbit in
five days. It could be further simplified according to the different
conditions of a spacecraft.

###  2.2 The significance of the pulse profiles

To get the ‘standard’ pulse profile, the arrival time of each photon on the
local site is corrected to the Solar System Barycenter (SSB) and folded with
the pulsar ephemeris with certain bins (Ge et al., 2012). In this process, the
orbit of the spacecraft plays an important role for the time correction. If
the orbit deviates from the correct position, the calculated profile will be
deformed from the ‘standard’ profile due to the wrong phase.

The significance of the pulse profile is defined as follows;

$\chi^{2}=\sum_{i=1}^{n}\frac{({P(\phi_{i})-\bar{P}})^{2}}{\bar{P}},$ (1)

where $P(\phi_{i})$ is the counts of the profile at $\phi_{i}$ , $\bar{P}$ is
the mean counts of the profile and $n$ is the total bin number of the profile.

It is expected that the significance of the calculated profile varies with
spacecraft orbit deviation. The more the orbit deviates from the real one, the
more the profile will be deformed, and the less the significance. $\chi^{2}$
will reach the maximum for the zero deviation, i.e., the true orbit.

We make a simulation for five-day observations of the Crab pulsar with HE. The
true orbit ‘ob0’ is given by the GPS on the satellite, and the orbit ‘ob1’ is
calculated with the dynamics of orbits by changing the ‘semi-major axis’ with
200 m. As shown in Fig. 1, the deviation will be larger, to more than 50 km
after 2 days. Using the predicted orbit ‘ob1’, the profile is deformed and the
significance is decreased from $1.237\times 10^{6}$ to $1.226\times 10^{6}$,
i.e. by about $1\%$.

Figure 1: The orbit deviation due to the change of initial orbit element
‘semi-major axis’ by 200 m and its effects on the profile. (a) orbit
deviations for 2 days; (b) zoom in of the orbit deviations. (c) distortion of
pulse profile induced by the change of initial element. The red line
represents the ‘standard’ profile generated with ‘ob0’, and the blue one
represents the distorted profile generated with ‘ob1’; (d) differences between
the ‘standard’ profile and distorted profile

###  2.3 The orbit fitting with grid search

Considering the correlations between the significance of the profile and the
orbit deviation, the orbit determination algorithm is described by the
processing flow in Fig. 2. In this algorithm, the orbit forecast and the
profile’s significance analysis are combined. The predicted orbit is obtained
with orbital dynamics model to complete the profile folding. At the same time,
the long-term stability characteristics of the profile can be used to
continuously update the orbital parameters, thus the long-term drift of the
orbit model can be corrected.

Figure 2: Processing flow of pulsar navigation.

On one hand, the orbit forecast model is constructed by considering the orbit
dynamics as described in Section 2.1. Then, with the estimated orbit
parameters $p_{0}$ ($a$, $e$, $i$, $\omega$, $\Omega$, $w$) at the current
time ($t_{0}$) and the certain parameter spaces $\delta p_{k}\,(k=1,2,..N)$,
the $k^{\rm th}$ predicted orbit is generated for the time span [$t_{s}$,
$t_{0}$] ($t_{s}$ is the beginning time for the used historic observation
data). On the other hand, the used data with the time span [$t_{s}$, $t_{0}$]
are chosen and filtered according to the selection criteria as described in
Section 3.

With the predicted orbit at different grid point, the arrival times of photons
are corrected to SSB and folded to generate the ‘predicted’ profiles using
phases calculated with the following equation (Ge et al., 2012);

$\Phi-\Phi_{0}=\nu{(t-t_{e})}+\frac{1}{2}\dot{\nu}{(t-t_{e})}^{2}+\frac{1}{6}\ddot{\nu}{(t-t_{e})}^{2},$
(2)

where $\Phi_{0}$, $\nu$, $\dot{\nu}$ and $\ddot{\nu}$ are the phase,
frequency, frequency derivative and the second order derivative of frequency
at the reference epoch $t_{e}$, respectively. Then, all the significances for
different profiles are calculated. Finally, the ‘significance analysis’ is
carried out, i.e. the trend between the significance of profiles and the orbit
parameters is fitted with the Gaussian function and the optimal orbit
parameters are obtained.

##  3 Data reduction

Almost five-day observation data of the Crab pulsar from 2017-08-31 10:00:00
to 2017-09-05 08:00:00 (UTC) are chosen, including all the data collected by
HE, ME, and LE. The data are analyzed with the _Insight_ -HXMT Data Analysis
software (HXMTDAS) software v2.0111http://www.uu-
world.cn/hxmt/hxmtsoft/index2.html. The following data selection criteria as
recommended in the software are used:

(1) Pointing offset angles $\leq 0.05^{\circ}$;

(2) Elevation angles $\geq 12^{\circ}$;

(3) Geomagnetic cutoff rigidity $\geq 6$;

(4) Only the detectors with small FoVs selected.

In addition to the above criteria, intervals with high background induced by
the charged particles or bright earth are excluded manually. Then we get the
filtered data with the total exposure duration of 180 ks, 190 ks, and 120 ks
for HE, ME, LE, respectively. The arrival times of the photons are corrected
to SSB, then the pulsed profiles are folded for HE, ME, and LE. The pulsar
ephemeris, shown in Table 1, is obtained from _Insight_ -HXMT data with the
timing residual of about 10 $\mu$s (1$\sigma$). As the $\ddot{\nu}$ parameter
of the Crab pulsar is about $10^{-20}\,{\rm s}^{-3}$, the affected pulsar
phase offset after 5 days is negligible ($\frac{1}{6}\cdot\ddot{\nu}\cdot
dt^{3}=1.3\times 10^{-4}$), thus we set $\ddot{\nu}=0$.

Table 1: The pulsar parameters of the Crab pulsar RA | $05^{\rm h}34^{\rm m}31^{\rm s}.973$
---|---
DEC | $22^{\circ}00^{\prime}52.06^{\prime\prime}$
$\nu$ | 29.639022542326
$\dot{\nu}$ | $-3.6867\times 10^{-10}$
$\ddot{\nu}$ | 0
EPOCH | 57992.16548081821

##  4 Results

The orbital elements of _Insight_ -HXMT are given at 07:59:00 (UTC) on Sept.
5th, 2017, and the space grids of parameters are chosen as listed in Table 2.
For an easy display, only the deviations of the parameters ($a$, $e$, $i$,
$\omega$, $\Omega$, $w$) from the true values ($6922.8781$ km, $0.00181017$,
$42.9715^{\circ}$, $207.0229^{\circ}$, $116.9049^{\circ}$, $22.5215^{\circ}$)
are listed.

Table 2: The orbit parameter spaces. ’$\Delta a$’, ’$\Delta e$’, ’$\Delta i$’, ’$\Delta\omega$’, ’$\Delta\Omega$’ and ’$\Delta w$’ are the deviations for semi-major axis, eccentricity, inclination angle, RAAN, argument of perigee, and mean anomaly, respectively. Orbital element deviation | Min | Max | Step
---|---|---|---
$\Delta$ a (m) | -200 | 200 | 20
$\Delta$ e | -0.0018 | 0.0182 | 0.001
$\Delta$ i (∘) | -0.5 | 0.5 | 0.05
$\Delta$ $\omega$ (∘) | -1.0 | 1.0 | 0.1
$\Delta$ $\Omega$ (∘) | -1.0 | 1.0 | 0.1
$\Delta$ w (∘) | -1.0 | 1.0 | 0.1

First, the orbit determination method is demonstrated with LE. At each grid
point of the parameter space, the ‘predicted’ profile is folded and its
significance is obtained. As shown in Fig. 3, the significance of the profile
varies significantly with the deviation of the orbital parameters and has a
maximum around the true value (zero deviation). Then the maxima obtained by
fitting a Gaussian function curve give the optimal values for the six orbital
parameters separately, i.e. each time one parameter is free with the other
five fixed, and the results are shown in Table 3. The process obtaining the
best parameters is very similar to that used by Sheikh et al to get the TOAs
from the observed pulse profiles (Sheikh, 2005). Therefore, the uncertainties
of these parameter values can be obtained in a similar way as follows;

$\sigma_{p}=\frac{\frac{1}{2}W}{SNR},$ (3)

where W is Full Width Half Maximum (FWHM), and SNR is the signal to noise
ratio obtained by the bootstrap method (Diaconis & Efron, 1983). As mentioned
by Sheikh (Sheikh, 2005), the SNR is limited to a maximum of 1000 by:

$SNR_{\rm filtered}=\frac{1000SNR}{1000+SNR}.$ (4)

Then similar demonstrations are also done for HE and ME, shown in Table 3. It
shows that the errors for HE are less than for ME and LE, because it has
collected more pulsed photons. Combining all the data from HE, ME and LE, the
errors become smaller. Transfer the orbit elements to the cartesian
parameters, we obtain the 3$\sigma$ errors of the current position and
velocity, which are 3.15 km, 6.61 km, and 4.46 km for x, y, z and 0.0073 km
s-1, 0.0033 km s-1, and 0.0042 km s-1 for $v_{x}$, $v_{y}$, $v_{z}$,
respectively. In short, the position and velocity are pinpointed within 10 km
(3$\sigma$) and 10 m s-1 (3$\sigma$).

Figure 3: Orbit determine result with Insight/LE: $\chi^{2}$ of the orbit elements of calculations (blue squares) and fitted results (red lines). ’$\Delta$a’, ’$\Delta$e’, ’$\Delta$i’, ’$\Delta$$\omega$’, ’$\Delta$$\Omega$’ and ’$\Delta{w}$’ are the deviations for semi-major axis, eccentricity, inclination angle, RAAN, argument of perigee, and mean anomaly, respectively. Table 3: Best estimated value of the elements and errors (3$\sigma$) Payload | Orbital element deviation |  $\Delta$a |  $\Delta$e |  $\Delta$i | $\Delta\omega$ | $\Delta\Omega$ |  $\Delta$w
---|---|---|---|---|---|---|---
|  | m | $10^{-3}$ | ∘ | ∘ | ∘ | ∘
LE | deviation | 7.02 | -0.38 | 0.01 | 0.01 | 0.003 | 0.003
| error ($3\sigma$) | 11.75 | 1.03 | 0.02 | 0.06 | 0.09 | 0.09
ME | deviation | 3.07 | -0.11 | 0.005 | 0.01 | 0.007 | 0.006
| error ($3\sigma$) | 16.44 | 1.28 | 0.03 | 0.064 | 0.11 | 0.11
HE | deviation | 4.89 | -0.58 | 0.006 | 0.009 | 0.002 | 0.002
| error ($3\sigma$) | 7.58 | 0.59 | 0.01 | 0.03 | 0.05 | 0.05
all detectors | deviation | 5.25 | -0.50 | 0.007 | 0.01 | 0.002 | 0.002
| error ($3\sigma$) | 6.40 | 0.52 | 0.01 | 0.03 | 0.04 | 0.04

##  5 Discussions

Conventionally, a minimum of three pulsars are required to get the absolute
navigation. Considering the clock time-offset on the spacecraft, another
pulsar is needed. Thus at least four pulsars should be observed
simultaneously, which would require at least four detectors, increasing the
technical complexity and the cost of mass, energy and money. An alternate
choice is to observe different pulsars sequentially using a single detector,
which would increase the risk of the control system. NICER has adopted the
latter strategy. With the large effective areas and low background rates, it
could get the TOAs of some millisecond pulsars in several kiloseconds. By
observing five millisecond pulsars sequentially, the spacecraft position was
determined under real flight conditions successfully (Alexandra, 2017).

The method SEPO, determining the orbit by using only one pulsar, has been
proposed and demonstrated with POLAR (Zheng et al., 2017) and _Insight_ -HXMT
in this paper, respectively. As POLAR has an effective area of about 200 cm2
and high background rates (about 4000 counts s-1), the orbit was determined
within 20 km (1 $\sigma$) by monitoring the Crab pulsar for 1 month. However,
it should be pointed out that the POLAR’s response for photons varies with
their incident angles, thus the obtained different profiles are mixed and
become ‘broader’, decreasing the navigation precision which is shown in
appendix A. For _Insight_ -HXMT, with the pointed observation mode, the
response to the incident pulsar emission is almost the same. With the
observations of the Crab pulsar for 5 days, the position is determined within
10 km. So it is feasible to use one detector for navigation by monitoring one
pulsar for a long time, which would result in low request of mass, energy, and
less spacecraft control. It is favorable for the navigation applications,
particular for the deep space navigation during the cruise phase of flight.

In addition, the standard pulse profile and parameters are important external
inputs in X-ray pulsar navigation. However, it is hard to achieve the fixed
‘standard’ profile in space X-ray detection. For example, if the energy band
or the response of the instrument is different, the ‘standard’ profiles become
different. Furthermore, the response of the instrument may vary in the long-
term operation in space, thus the profile varies too. If we still use the
previous ‘standard’ profile, it would inevitably affect the navigation
accuracy and even result in unpredictable systematic deviation. For SEPO,
neither the standard pulse profile nor the continuous update of the pulsar
parameters is needed. Therefore, it is no longer dependent on the standard
pulse profile, nor affected by the pulsar observation conditions and the
detector performance changes. It can be applied to different detectors and
different mission scenarios.

For the in-orbit detections of a pulsar, the obtained profile could be
regarded as the summation of all the profiles at different point of the orbit.
That is,
$P(\phi_{i})=\sum_{k=1}^{m}p_{k}(\phi_{i})=\sum_{k=1}^{m}p(\phi_{i}-\delta{\phi_{k}})$,
where $\delta{\phi_{k}}$ represents the phase offset due to the position
deviation. Thus, considering the significance of the profile defined in Sec.
2.2,

$\displaystyle\chi^{2}$
$\displaystyle=\sum_{i=1}^{n}\\{{\sum_{k=1}^{m}p(\phi_{i}-\delta{\phi_{k}})-\bar{P}}\\}^{2}/\bar{P}$
(5)
$\displaystyle=\sum_{i=1}^{n}\sum_{k=1}^{m}\sum_{k^{\prime}=1}^{m}p(\phi_{i}-\delta{\phi_{k}})p(\phi_{i}-\delta{\phi_{k^{\prime}}})/\bar{P}-n\bar{P}.$

As shown in equation 5, $\chi^{2}$ of the profile will reach the maximum with
$\delta\phi_{k}=\delta\phi_{k^{\prime}}$, i.e. $\phi_{k}=const$. $\phi_{k}=0$
represents the true orbit with zero deviation. If $\phi_{k}=const$ (nonzero),
it means that the orbit has fixed deviations from the real orbit. However,
constrained by the orbital dynamics, there is few such completely ‘parallel’
tracks, thus ensuring to find the true orbit. For the cruising orbit in deep
space, there indeed exist approximately parallel tracks. In such conditions,
more measurements besides the significance of the profiles, TOAs for example,
are needed to discriminate the true orbit from the others.

The proposed SEPO method, however, has not been proven mathematically. Thus
some simulations have been performed and the detailed results are shown in the
appendix, which shows that the method works with very different pulse
profiles. In this in-orbit demonstration with the SEPO method, the position is
pinpointed to within 10 km (3$\sigma$), while the theoretical bound is $\sim$
250 meters (3$\sigma$) for 1920 m${}^{2}\cdot$s of LE (the effective area of
160 cm2 for the small FOV with the exposure duration of 120 ks) (Hanson et al,
2008). Possible reason are the effects of orbit forecast error ($<3.5$ km),
absolute time accuracy of _Insight_ -HXMT (Li et al., 2018), or the intrinsic
timing noise of the Crab pulsar. The ephemeris used in this demonstration has
an timing residual of $\sim$10 $\mu$s (1$\sigma$), resulting in position error
of $\sim$9.0 km (3$\sigma$). See the detailed discussions on the effects of
timing noise at the end of this section.

By now, the clock time-offset on the spacecraft has not been considered in the
SEPO method. The short-term drift of clock within the integration time would
deform the calculated profile, change the pulse significance and result in
additional systematic errors. While the long-term drift just changes the
absolute TOA, which has little effects on the pulse significance. Thus the
clock time-offset can be taken into account and corrected with the
significance analysis in future. In addition, the pointing error also
decreases the significance of the profiles by changing the effective areas.
However, the pointing error has no effects on the pulse phase of the pulsar,
thus has little effect on the solution in the SEPO method, even if the
pointing error has a dependence on orbit position. In this work, the
observation of _Insight_ -HXMT was performed in the ‘pointed mode’ with the
pointing error $\leq 0.028^{\circ}$ (3$\sigma$). In this situation, the
effective area changes by only about 0.5%, which can be ignored completely.

The timing noise of pulsars (Hobbs et al, 2010) has a great influence on the
accuracy of pulsar navigation as shown above. The timing noise of a pulsar
includes white noise component and red noise component. White noise can be
suppressed by accumulating observation data over a long period of time.
However, the red noise is difficult to suppress or eliminate. Actually, the
effects of the red noise is quite similar to the clock time-offset and can be
corrected as mentioned above. On the other hand, combining with the study of
the mechanism of noise (such as magnetic field evolution), it is possible to
give a more accurate prediction of the spin evolutions of pulsars, reducing
the impact of timing noise (especially red noise) on the accuracy of pulsar
navigation (Yi & Zhang, 2015; Gao et al, 2016; Zhang & Xie, 2012a, b). In
addition, the glitch, that is, a sudden change in the rotation period, will
also have to be considered in future.

##  6 Conclusion

A new navigation method SEPO has been proposed that combines the observed
pulse profile with orbit dynamics. It has been demonstrated with the _Insight_
-HXMT observations of only one pulsar (Crab) and the orbit has been determined
successfully. Combining all the data, we obtain the best position estimations
within 10 km (3$\sigma$) and velocity within 10 m s-1 (3$\sigma$),
respectively.

This work is supported by the National Key R&D Program of China
(2016YFA0400800) and the National Natural Science Foundation of China under
grants U1838101 and 11503027. This work made use of the data from the
_Insight_ -HXMT mission, a project funded by China National Space
Administration (CNSA) and the Chinese Academy of Sciences (CAS). Appendix

##  Appendix A Simulations on the SEPO method

We perform some simulations to evaluate the feasibility of the new navigation
method SEPO, especially for different pulse-profiles. The duty cycle of the
profile $\delta$ is defined as follows;

$\delta=\frac{w}{d},$ (A1)

where $w$ is the width of the pulse (above 10% maximum of the peak), $d$ is
the pulsar period. Thus $\delta$ describes the sharpness of a pulse profile.

First, three different pulse profiles from three pulsars, i.e., PSRs B1821-24,
B1509-58, J1811-1925 and a supposed sinusoidal-like pulse shape are chosen for
test, as shown in Fig. 4 (the left panel). To exclude the effects of other
factors, the parameters such as position and period are fixed to be the same
as the Crab pulsar. The pulse count rate is assumed to be 130 cnts$\cdot$ s-1
and the background count rate to be 1200 cnts$\cdot$ s-1, as obtained from the
LE data. Then the _Insight_ -HXMT/LE observation data of the four pulsars are
simulated with the true orbit of _Insight_ -HXMT from Aug. 30th to Sept. 5th,
2019.

Then the method SEPO is carried out and the significance of the profiles with
the deviations of the elements (semi-major axis for example) are shown in Fig.
4 (the right panel). It shows that the significance of all the four different
pulse-profiles varies with $\Delta$a, and the optimal value can be found by
Gaussian-fitting, which is near the true value (zero). The pulse-profiles do
have effects on the position error. The sharper is the pulse-profile, the
larger is the calculated significance, and the smaller is the position error.
The duty cycle $\delta$ is 0.16, 0.55, 0.72, 0.80 for PSRs B1821-24, B1509-58,
J1811-1925 and sinusoidal-like pulsar. The profiles like PSR B1821-24 are the
best candidate due to its the navigation for the small duty cycle. For PSR
J1811-1925, the significance trend is not smooth, however, the optimal value
can still be obtained by fitting.

Figure 4: Left: the four pulse-profiles used in the simulations, ‘$\delta$’
represents the duty cycle as defined in the text; right: $\Delta\chi^{2}$ of
the deviations for semi-major axis ($\Delta$a) of calculations (black squares)
and fitted results (red lines). For clarity, $\Delta\chi^{2}$
($\chi^{2}-\chi^{2}_{\rm min}$) is shown. $\chi^{2}_{\rm min}$ is different
for different pulse-profiles, which is $7.09\times 10^{6}$, $1.00\times
10^{6}$, $7.22\times 10^{5}$, $5.05\times 10^{5}$ for PSRs B1821-24, B1509-58,
J1811-1925 and sin-like profile, respectively.

To estimate the effects of pulse shapes in details, a double-peak profile is
generated with $\delta=0.16$, and $\delta$ is increased by 1%, 5%, and 10%,
respectively. Then the observation data are simulated and the method SEPO is
carried out as described above. The results are shown in Fig. 5, and the
optimal semi-major axis and the errors ($3\sigma$) are shown in Table 4. It
also shows that the significance decreases as the profile gets broader. With
$\delta$ becoming larger by 10%, the resulted error of the semi-major axis is
increased by 8.8%.

Figure 5: $\Delta\chi^{2}$ of the deviations for semi-major axis ($\Delta$a) of calculations (black squares) and fitted results (red lines). For clarity, $\Delta\chi^{2}$ ($\chi^{2}-\chi^{2}_{\rm min}$) is shown. $\chi^{2}_{\rm min}$ is different for different pulse-profiles, which is $6.40\times 10^{6}$, $6.33\times 10^{6}$, $6.06\times 10^{6}$, $5.75\times 10^{6}$ for the changed profile with $\delta$ increased by 0%, 1%, 5%, and 10%, respectively.  Table 4: Best estimated value of $\Delta$a and errors (3$\sigma$) enlargement of $\delta$ |  | 0% | 1% | 5% | 10%
---|---|---|---|---|---
$\Delta$a (m) | deviation | -3.15 | -6.12 | -5.4 | -6.8
| error ($3\sigma$) | 24.96 | 25.12 | 26.12 | 27.15

## References

  * Hewish et al. (1968) Hewish, A., Bell, S. J., Pilkington, J. D. H., et al. 1968, Nature, 217, 709
  * Taylor (1991) Taylor, J. H. 1991, IEEEP 79,1054
  * Downs (1974) Downs, G. S. 1974, NASA Technical Report 32
  * Chester & Butman (1981) Chester, T. J., Butman, S. A. 1981, TDAPR, 63, 22-25.
  * Becker et al. (2013) Becker, W., Bernhardt, M. G., Jessner, A., 2013, AcFut, 7, 11
  * Wang et al. (2014) Wang, Y. D., Zheng, W., Sun, S. M., Li, L., 2014, Aerosp. Sci. Technol, 36, 27
  * Emadzadeh & Speyer (2011) Emadzadeh, A. A., Speyer, J. L., 2011, ITAES, 47,2317
  * Shemar et al. (2016) Shemar, S., Fraser, G., Heil, L., et al. 2016, ExA, 42, 101-138
  * Cui et al. (2016) Cui, P. Y., Wang, S., Gao, A., et al. 2016, AdSpR, 57,1889
  * Wei et al. (2013) Wei, E. H., Jin, S. G., Zhang, Q., et al. 2013, AdSpR, 51, 849.
  * Wood (1993) Wood, K. S., 1993, SPIE, 1940, 105
  * Sheikh (2005) Sheikh, S. I., 2005, PHD dissertation. University of Maryland (USA)
  * Zheng et al. (2017) Zheng, S. J., Ge, M. Y., Han, D. W. et al. 2017, SSPMA, 47, 9505
  * Produit et al. (2005) Produit, N., Barao, F., Deluit, S., et al. 2005, NIMPA, 550, 616
  * Li et al. (2018) Li, Z. H., Kole, M., Sun, J. C., et al. 2018, NIMPA, 900, 8
  * Paul et al. (2017) Paul, S. R., Kent, S. W., Michael, T. W., 2017, arXiv:1711.08507
  * Alexandra (2017) Alexandra, W., 2017, Nature News, 47, 9505
  * Zhang et al. (2018) Zhang, S., Zhang, S. N., Lu, F. J., et al. 2018, SPIE 10699, 1
  * Li et al. (2018) Li, X. B., Song, L. M., Li, X. F., et al. 2018, SPIE 10699, 69
  * Wang et al. (2018) Wang, Y. D., Zhang, D. P., Zheng, W., et al. 2018, ITAES, submitted
  * Wang & Gao (2016) Wang, W. B., Gao, Y., 2016, Trans. JSASS Aerospace Tech. 14, 39
  * Ge et al. (2012) Ge, M. Y., Lu, F. J., Qu, J. Lu., et al. 2012, ApJ, 199, 32
  * Diaconis & Efron (1983) Diaconis, P., & Efron, B., 1983, SciAm, 248,116
  * Hobbs et al (2010) Hobbs, G., Lyne, A. G., & Kramer, M., 2010, MNRAS, 402, 1027
  * Sheikh et al (2007) Sheikh, S. I., Golshan, A. R., and Pines, D. J., 2007 30th Annual AAS Guidance and Control Conference, 855-874
  * Hanson et al (2008) Hanson, J., Sheikh, S., Graven, P., and Collins, J., 2008 IEEE-ION Position Location and Navigation Symposium (PLANS), 704-713
  * Yi & Zhang (2015) Yi, S. X., Zhang, S. N., 2015, MNRAS, 454, 3674
  * Gao et al (2016) Gao, X. D., Zhang, S. N., Yi, S. X., et al. 2016, MNRAS, 459, 402
  * Zhang & Xie (2012a) Zhang, S. N., Xie Y., 2012, ApJ, 757, 153
  * Zhang & Xie (2012b) Zhang, S. N., Xie, Y., 2012, ApJ, 761, 102