Autonomous space navigation is a rapidly advancing technology in the space industry. One heavily researched area in space navigation relies on the directional sensing of X-ray pulsar rotations. Cramér–Rao lower bound (CRLB) is a statistical estimation tool widely used to calculate the variance of unbiased estimators such as the rotation parameters of a pulsar. An experimental analysis based on X-ray pulsar observation conducted by Professor Liang Zhao and colleagues offers a new CRLB for pulsar rotation parameters estimation. The study was conducted at the Science and Technology on Space Physics Laboratory in Beijing and a pre-proof report is available on Advances in Space Research.
What are Pulsars?
A neutron star spinning at a rapid pace is called a pulsar. The small, highly dense remnants left behind by a much bigger star is known as a neutron star. Due to the immense density of a neutron star, it weighs much more than the sun, although it is only a fraction of its size.
Strong magnetic fields are created within neutron stars due to their fast rotation. Such neutron stars produce high-energy beams at their north and south magnetic poles. Light pulses are observed when these beams are directed toward Earth and flash as the neutron star rotates. This observation was named pulsars by astronomers. The majority of neutron stars detected on earth are pulsars.
When there is a misalignment between the axis of rotation and the magnetic poles, X-rays are emitted from the pulsars. These X-ray pulsars exhibit various pulse shapes that differ from source to source. Different X-ray detection technologies, like X-ray telescopes, have been developed to observe X-ray pulsars. These measurements are used to map out the origin of the pulsars.
X-ray Pulsar-Based Navigation and Timing (XPNAV)
In X-ray pulsar-based navigation and timing (XPNAV), also known as "pulsar navigation," a vehicle, such as a spacecraft, uses the periodic X-ray signals released from pulsars to establish its location in deep space. Receiving X-ray signals would be compared by a vehicle utilizing XPNAV to a database of previously recorded pulsar information. This comparison would enable the vehicle to calculate its position much like GPS. But for the XPNAV to be efficient, it has to have accurate measurements of the frequency and timing of the pulsar rotation parameters.
The data required for navigation is acquired by the difference in the pulse's time of arrival (TOA). TOA is obtained by gathering X-ray photon signals and formulating a timing model.
The performance of pulsar navigation depends on the accuracy of the pulsar rotation frequency, which changes with the prediction accuracy of the pulse TOA.
To show the theoretical lower bound of the predicted variance, it is crucial to evaluate the correctness of the estimation. Theoretically, the estimator's lower bound is typically determined using the Cramér-Rao Lower Bound (CRLB). The CRLB is a mathematical tool that was developed for frequency estimation.
Developing the Cramér-Rao Lower Bound (CRLB) Model
Professor Zhao’s group derived the CRLB for pulsar rotation parameters estimate with X-ray pulsar observation data to serve as a benchmark for identifying the least variance of frequency and its derivatives estimation.
Their work is an analytical expression for the first and second derivatives of the pulsar frequency and its CRLB. The calculated CRLB is also checked for accuracy by comparing it to the root mean square error (RMSE) in simulation experiments.
The estimation methods integrated the measured photon TOA to approximate the pulsar frequency and its all-order derivatives. Through repeated experiments, several sets of data were collected. Subsequently, the RMSE of the estimated timing model parameter was computed and compared to the derived CRLB.
Ground test systems were used to mimic X-ray pulsar transmissions under various circumstances to verify the theoretical model.
According to the experimental findings, the RMSE steadily approaches the estimated CRLB as the observation duration grows. In 2.4 106 s, the difference between the RMSE of pulsar frequency estimates and the CRLB remains at 10–11 orders of magnitude. This demonstrates that the lower theoretical bound for predicting pulsar rotation parameters, the CRLB expression derived in this study, is correct. By serving as a comparison point between the minimal variance estimator and other estimators, the resulting CRLB in this study can be used in the future for pulsar rotation parameter estimation.
The results demonstrate that the estimated lower bound for pulsar rotation parameters is the mathematical expression of the determined CRLB. Furthermore, by serving as a comparison point between the minimal variance estimator and other estimators, the deduced CRLB in this study aids in developing the minimum variance estimator for pulsar rotation parameter estimation using X-ray pulsar data.
Jianyu Su, Haiyan Fang, Weimin Bao, Haifeng Sun, Liang Zhao. Cramér-Rao Lower Bound for Pulsar Rotation Parameters Estimation with X-ray Pulsar Observation Data. Advances in Space Research, 2022, ISSN 0273-1177, www.sciencedirect.com/science/article/abs/pii/S0273117722008407?via%3Dihub.