Solar System Ephemeris Research Articles

Excluding Primordial Black Holes as Dark Matter Based on Solar System Ephemeris

Current cosmological constraints allow primordial black holes to constitute dark matter in the mass range of 1018–1022 g. I show that a major portion of this logarithmic window can be ruled-out based on the solar system ephemeris, given that the external mass enclosed within 50 au from the Sun did not change by more than ∼5 × 10−14 M ⊙ yr−1 in recent decades.

The effect of the Solar System Ephemeris on the search for the nano-Hz gravitational wave background

ABSTRACT The detection of the nano-Hz gravitational-wave background (GWB) is one of the main targets of Pulsar timing arrays (PTAs). The detection can be achieved via searching for a common signal with quadrapolar correlation between pulsar pairs. Errors in the Solar-System ephemeris (SSE) can induce dipolar correlations in PTA data, which may affect the results of GWB searches, especially when the data quality is not high enough to constrain the correlation pattern. We investigate the effect of unmodelled SSE errors on GWB searches with PTAs, using simulations with properties based on the European Pulsar Timing Array data set. When the GWB signal is strong, SSE errors have little effect on the GWB search results, including parameter inference and model selection. When the GWB signal is weak, SSE errors can lead to overestimation of the GWB amplitude. However, model comparison would show strong support for dipolar correction, which implies the source of the common signal to be SSE-related and helps avoid its misidentification as a GWB signal. This indicates that SSE error is unlikely to be the main source of the common signal detected recently with real PTA data. We also use simulations to test the ability of the SSE model LINIMOSS in absorbing SSE-error signals and leaving the GWB signal intact. We show that marginalizing LINIMOSS planetary parameters with infinite priors is good at absorbing SSE errors, but may also lead to improper absorption of a GWB signal. Caution is therefore required when setting appropriate limits on the priors of SSE parameters.

Celestial Mechanics Leading up to the AAS Division on Dynamical Astronomy

This talk reviewed key developments leading to the establishment of the AAS Division on Dynamical Astronomy. In the 1890s, Simon Newcomb calculated astronomical constants and ephemerides for solar system bodies that were internationally adopted and used for 80 yr. The introduction of punch card equipment in the 1940s and computers in the 50s allowed much faster computations. Launches of Sputnik, Explorer, Vanguard, and Mercury satellites led to computing the orbits of satellites around the Earth and the field of astrodynamics. Dirk Brouwer's death in 1966 resulted in the celestial mechanics faculty at Yale spreading across the country. The space program, radar observations, and faster computers propelled advances in celestial mechanics. Teams at the Jet Propulsion Laboratory and Massachusetts Institute of Technology developed solar system ephemerides using numerical integrations fit to optical observations and radar ranging measurements. It was time for the AAS Division on Dynamical Astronomy.

The Gravitational-wave Background Null Hypothesis: Characterizing Noise in Millisecond Pulsar Arrival Times with the Parkes Pulsar Timing Array

The noise in millisecond pulsar (MSP) timing data can include contributions from observing instruments, the interstellar medium, the solar wind, solar system ephemeris errors, and the pulsars themselves. The noise environment must be accurately characterized in order to form the null hypothesis from which signal models can be compared, including the signature induced by nanohertz-frequency gravitational waves (GWs). Here we describe the noise models developed for each of the MSPs in the Parkes Pulsar Timing Array (PPTA) third data release, which have been used as the basis of a search for the isotropic stochastic GW background. We model pulsar spin noise, dispersion measure variations, scattering variations, events in the pulsar magnetospheres, solar wind variability, and instrumental effects. We also search for new timing model parameters and detected Shapiro delays in PSR J0614−3329 and PSR J1902−5105. The noise and timing models are validated by testing the normalized and whitened timing residuals for Gaussianity and residual correlations with time. We demonstrate that the choice of noise models significantly affects the inferred properties of a common-spectrum process. Using our detailed models, the recovered common-spectrum noise in the PPTA is consistent with a power law with a spectral index of γ = 13/3, the value predicted for a stochastic GW background from a population of supermassive black hole binaries driven solely by GW emission.

The photometric observation of the quasi-simultaneous mutual eclipse and occultation between Europa and Ganymede on 22 August 2021

Mutual events (MEs) are eclipses and occultations among planetary natural satellites. Most of the time, eclipses and occultations occur separately. However, the same satellite pair will exhibit an eclipse and an occultation quasi-simultaneously under particular orbital configurations. This kind of rare event is termed as a quasi-simultaneous mutual event (“QSME”). During the 2021 campaign of mutual events of jovian satellites, we observed a QSME between Europa and Ganymede. The present study aims to describe and study the event in detail. We observed the QSME with a CCD camera attached to a 300-mm telescope at the Hong Kong Space Museum Sai Kung iObservatory. We obtained the combined flux of Europa and Ganymede from aperture photometry. A geometric model was developed to explain the light curve observed. Our results are compared with theoretical predictions (“O-C”). We found that our simple geometric model can explain the QSME fairly accurately, and the QSME light curve is a superposition of the light curves of an eclipse and an occultation. Notably, the observed flux drops are within 2.6% of the theoretical predictions. The size of the event central time O-C’s ranges from −14.4 to 43.2 s. Both O-C’s of flux drop and timing are comparable to other studies adopting more complicated models. Given the event rarity, model simplicity and accuracy, we encourage more observations and analysis on QSMEs to improve Solar System ephemerides.

主小行星带对火星轨道长期动力学影响评估

<p indent="0mm">The successful implementation of the Tianwen-1 mission marks that China is moving toward a deeper space beyond the moon, facing a more complex space dynamics environment, and the difficulty of deep space exploration will continue to increase. The main belt asteroids (MBAs) are adjacent to Mars, and the perturbation by MBAs is hard to fully consider, effecting the accuracy of the Mars orbital dynamics parameters. Thus, it will indirectly affect the construction of a high-precision solar system ephemeris and the Mars exploration missions. This paper analyzes the distribution feature of MBAs based on the Nesvorný asteroid library. The dynamic environment of the whole solar system with MBAs was established to evaluate MBA’s perturbation influence. The results show that the difference between the ten-year integral orbits of the inner solar system planets and DE421 is less than <sc>15 m,</sc> and the integral difference of Mars’ orbits reaches<sc>100 m.</sc> Based on the point-mass model, the one-ring model and the latest six-ring model, the influence of MBAs on the orbit of Mars is estimated, and the orbit error of <sc>10 years</sc> is about <sc>2 km,</sc><sc> 31.3 m,</sc> and <sc>2 m,</sc> respectively. Further, the joint solution of the ring model and the orbital parameters of the planets is discussed, which confirms that the residuals of Mars-Earth distance with the fitted ring model can theoretically be reduced to sub-meters. The error in the <italic>Z</italic>-axis of the planet’s orbit is large if use only the range measurement.

Effect of Ephemeris on Pulsar Timing and Navigation Accuracy Based on X-ray Pulsar Navigation-I Data

Solar system ephemeris is very important for pulsar timing and navigation. In order to explore the effect of different precision ephemerides on X-ray pulsar timing and navigation, the differences between timing and navigation results with four JPL Development Ephemerides based on the data of X-ray pulsar navigation-I (XPNAV-I) were compared and analyzed in this paper. For pulsar timing, the ephemeris has a systematic effect on time scale conversion (nanosecond difference), light-travel delay (millisecond difference) and timing residuals (microsecond difference), and the pulse profile phase can reflect the systematic deviation caused by different ephemerides in the timing calculation. The timing results show that it is necessary to compile the pulsar timing model based on the newer ephemeris. For navigation, based on the significant enhancement of pulse profile with orbit-dynamic (SEPO), the absolute error between simulation orbit and actual orbit is less than 2 km for each ephemeris, and the differences between simulation orbits are less than 1 km. The orbit position accuracy calculated by the ephemeris used in pulsar timing parameter calculation is the highest (DE200 in this paper), which explains the necessity of using a unified ephemeris in the calculation of timing and navigation with satisfying its internal self-consistency.

The NANOGrav 12.5-year Data Set: Search for Non-Einsteinian Polarization Modes in the Gravitational-wave Background

Abstract We search NANOGrav’s 12.5 yr data set for evidence of a gravitational-wave background (GWB) with all the spatial correlations allowed by general metric theories of gravity. We find no substantial evidence in favor of the existence of such correlations in our data. We find that scalar-transverse (ST) correlations yield signal-to-noise ratios and Bayes factors that are higher than quadrupolar (tensor-transverse, TT) correlations. Specifically, we find ST correlations with a signal-to-noise ratio of 2.8 that are preferred over TT correlations (Hellings and Downs correlations) with Bayesian odds of about 20:1. However, the significance of ST correlations is reduced dramatically when we include modeling of the solar system ephemeris systematics and/or remove pulsar J0030+0451 entirely from consideration. Even taking the nominal signal-to-noise ratios at face value, analyses of simulated data sets show that such values are not extremely unlikely to be observed in cases where only the usual TT modes are present in the GWB. In the absence of a detection of any polarization mode of gravity, we place upper limits on their amplitudes for a spectral index of γ = 5 and a reference frequency of f yr = 1 yr−1. Among the upper limits for eight general families of metric theories of gravity, we find the values of A TT 95 % = ( 9.7 ± 0.4 ) × 10 − 16 and A ST 95 % = ( 1.4 ± 0.03 ) × 10 − 15 for the family of metric spacetime theories that contain both TT and ST modes.

Common-red-signal analysis with 24-yr high-precision timing of the European Pulsar Timing Array: inferences in the stochastic gravitational-wave background search

ABSTRACT We present results from the search for a stochastic gravitational-wave background (GWB) as predicted by the theory of General Relativity using six radio millisecond pulsars from the Data Release 2 (DR2) of the European Pulsar Timing Array (EPTA) covering a timespan up to 24 yr. A GWB manifests itself as a long-term low-frequency stochastic signal common to all pulsars, a common red signal (CRS), with the characteristic Hellings-Downs (HD) spatial correlation. Our analysis is performed with two independent pipelines, ENTERPRISE, and TEMPONEST+FORTYTWO, which produce consistent results. A search for a CRS with simultaneous estimation of its spatial correlations yields spectral properties compatible with theoretical GWB predictions, but does not result in the required measurement of the HD correlation, as required for GWB detection. Further Bayesian model comparison between different types of CRSs, including a GWB, finds the most favoured model to be the common uncorrelated red noise described by a power law with $A = 5.13_{-2.73}^{+4.20} \times 10^{-15}$ and $\gamma = 3.78_{-0.59}^{+0.69}$ (95 per cent credible regions). Fixing the spectral index to γ = 13/3 as expected from the GWB by circular, inspiralling supermassive black hole binaries results in an amplitude of $A =2.95_{-0.72}^{+0.89} \times 10^{-15}$. We implement three different models, BAYESEPHEM, LINIMOSS, and EPHEMGP, to address possible Solar system ephemeris (SSE) systematics and conclude that our results may only marginally depend on these effects. This work builds on the methods and models from the studies on the EPTA DR1. We show that under the same analysis framework the results remain consistent after the data set extension.

The NANOGrav 12.5 yr Data Set: Search for an Isotropic Stochastic Gravitational-wave Background

Abstract We search for an isotropic stochastic gravitational-wave background (GWB) in the 12.5 yr pulsar-timing data set collected by the North American Nanohertz Observatory for Gravitational Waves. Our analysis finds strong evidence of a stochastic process, modeled as a power law, with common amplitude and spectral slope across pulsars. Under our fiducial model, the Bayesian posterior of the amplitude for an f −2/3 power-law spectrum, expressed as the characteristic GW strain, has median 1.92 × 10−15 and 5%–95% quantiles of 1.37–2.67 × 10−15 at a reference frequency of f yr = 1 yr − 1 ; the Bayes factor in favor of the common-spectrum process versus independent red-noise processes in each pulsar exceeds 10,000. However, we find no statistically significant evidence that this process has quadrupolar spatial correlations, which we would consider necessary to claim a GWB detection consistent with general relativity. We find that the process has neither monopolar nor dipolar correlations, which may arise from, for example, reference clock or solar system ephemeris systematics, respectively. The amplitude posterior has significant support above previously reported upper limits; we explain this in terms of the Bayesian priors assumed for intrinsic pulsar red noise. We examine potential implications for the supermassive black hole binary population under the hypothesis that the signal is indeed astrophysical in nature.

Gravitational wave sensitivity curve of pulsar timing arrays affected by correlated noises

The detection of gravitational waves (GWs) by pulsar timing arrays (PTAs) is not only a very important supplementation of the verification of general relativity but also a new window to study the evolution of supermassive binary black holes and the early universe. However, so far the detection sensitivity of PTAs is not good enough to catch signals of GWs due to the disturbance of various noises. In this paper we explore the influences of the correlated noises caused by the reference clock errors and solar system ephemeris errors in pulsar timing on the detection of stochastic gravitational waves background (GWB). We demonstrate the power-law integrated sensitivity curves of GWB detection under the impacts from these correlated noises. From the simulated data, we find that the influence of different reference time-scale is non-negligible, and the influence is even quite huge if one uses a very old version of solar system ephemeris. However, the impact from these correlated noises on the sensitivity curve is very limited for the real observational data released by international pulsar timing arrays (IPTA). We also calculate the signal-to-noise ratios based on the theoretical GW amplitude permitted by observations. Moreover, we study how the detection sensitivity increases with more pulsar number and longer observation.

Revisiting profile instability of PSR J1022+1001

ABSTRACT Millisecond pulsars in timing arrays can act as probes for gravitational wave detection and improving the Solar system ephemerides among several other applications. However, the stability of the integrated pulse profiles can limit the precision of the ephemeris parameters and in turn the applications derived from it. It is thus crucial for the pulsars in the array to have stable integrated pulse profiles. Here we present evidence for long-term profile instability in PSR J1022+1001 which is currently included in the European and Parkes pulsar timing arrays. We apply a new evaluation method to an expanded data set ranging from the Effelsberg Pulsar Observing System backend used in the 1990s to that of data from the current PSRIX backend at the Effelsberg Radio Telescope. We show that this intrinsic variability in the pulse shape persists over time-scales of years. We investigate if systematic instrumental effects like polarization calibration or signal propagation effects in the interstellar medium causes the observed profile instability. We find that the total variation cannot be fully accounted for by instrumental and propagation effects. This suggests additional intrinsic effects as the origin for the variation. We finally discuss several factors that could lead to the observed behaviour and comment on the consequent implications.

Search for ultralight scalar dark matter with NANOGrav pulsar timing arrays

An ultralight scalar field is a candidate for the dark matter. The ultralight scalar dark matter with mass around 10−23 eV induces oscillations of the pulse arrival time in the sensitive frequency range of the pulsar timing arrays. We search for the ultralight scalar dark matter using the North American Nanohertz Observatory for Gravitational Waves 11-year Data Set. We give the 95% confidence upper limit for the signal induced by the ultralight scalar dark matter. In comparison with the published Bayesian upper limits on the amplitude of the ultralight scalar dark matter obtained by Bayesian analysis using the Parkes Pulsar Timing Array 12-year data set (Porayko et al. 2018), we find three times stronger upper limit in the frequency range from 10−8.34 to 10−8.19 Hz which corresponds to the mass range from 9.45×10−24 to 1.34×10−23 eV. In terms of the energy density of the dark matter, we find that the energy density near the Earth is less than 7 GeV/cm3 in the range from 10−8.55 to 10−8.01 Hz (from 5.83×10−24 to 2.02×10−23 eV). The strongest upper limit on the energy density is given by 2 GeV/cm3 at a frequency 10−8.28 Hz (corresponding to a mass 1.09×10−23 eV). We find that the signal of the ultralight scalar dark matter is strongly covariant with the solar system ephemeris effect. Also, we reveal that the model of the solar system ephemeris effect prefers parameters which are contrary to the expectation that noise will be reduced on all pulsars.

Modeling the Uncertainties of Solar System Ephemerides for Robust Gravitational-wave Searches with Pulsar-timing Arrays

Abstract The regularity of pulsar emissions becomes apparent once we reference the pulses’ times of arrivals to the inertial rest frame of the solar system. It follows that errors in the determination of Earth's position with respect to the solar system barycenter can appear as a time-correlated bias in pulsar-timing residual time series, affecting the searches for low-frequency gravitational waves performed with pulsar-timing arrays. Indeed, recent array data sets yield different gravitational-wave background upper limits and detection statistics when analyzed with different solar system ephemerides. Crucially, the ephemerides do not generally provide usable error representations. In this article, we describe the motivation, construction, and application of a physical model of solar system ephemeris uncertainties, which focuses on the degrees of freedom (Jupiter's orbital elements) most relevant to gravitational-wave searches with pulsar-timing arrays. This model, BayesEphem, was used to derive ephemeris-robust results in NANOGrav's 11 yr stochastic-background search, and it provides a foundation for future searches by NANOGrav and other consortia. The analysis and simulations reported here suggest that ephemeris modeling reduces the gravitational-wave sensitivity of the 11 yr data set and that this degeneracy will vanish with improved ephemerides and with pulsar-timing data sets that extend well beyond a single Jovian orbital period.

Constraints on individual supermassive binary black holes using observations of PSR J1909–3744

We perform a search for gravitational waves (GWs) from several supermassive binary black hole (SMBBH) candidates (NGC 5548, Mrk 231, OJ 287, PG 1302–102, NGC 4151, Ark 120 and 3C 66B) in long-term timing observations of the pulsar PSR J1909–3744 obtained using the Parkes radio telescope. No statistically significant signals were found. We constrain the chirp masses of those SMBBH candidates and find the chirp mass of NGC 5548 and 3C 66B to be less than 2.4 × 109 M⊙ and 2.5 × 109 M⊙ (with 95% confidence), respectively. Our upper limits remain a factor of 3 to 370 above the likely chirp masses for these candidates as estimated from other approaches. The observations processed here provide upper limits on the GW strain amplitude that improve upon the results from the first Parkes Pulsar Timing Array data release by a factor of 2 to 7. We investigate how information about the orbital parameters can help to improve the search sensitivity for individual SMBBH systems. Finally, we show that these limits are insensitive to uncertainties in the Solar System ephemeris model.

Studying the Solar system dynamics using pulsar timing arrays and the LINIMOSS dynamical model

ABSTRACT Pulsar timing arrays (PTAs) can be used to study the Solar system ephemeris (SSE), the errors of which can lead to correlated timing residuals and significantly contribute to the PTA noise budget. Most Solar system studies with PTAs assume the dominance of the term from the shift of the Solar system barycentre (SSB). However, it is unclear to which extent this approximation can be valid, since the perturbations on the planetary orbits may become important as data precision keeps increasing. To better understand the effects of SSE uncertainties on pulsar timing, we develop the linimoss dynamical model of the Solar system, based on the SSE of Guangyu Li. Using the same input parameters as DE435, the calculated planetary positions by linimoss are compatible with DE435 at centimetre level over a 20 yr timespan, which is sufficiently precise for pulsar-timing applications. We utilize linimoss to investigate the effects of SSE errors on pulsar timing in a fully dynamical way, by perturbing one SSE parameter per trial and examining the induced timing residuals. For the outer planets, the timing residuals are dominated by the SSB shift, as assumed in previous work. For the inner planets, the variations in the orbit of the Earth are more prominent, making previously adopted assumptions insufficient. The power spectra of the timing residuals have complex structures, which may introduce false signals in the search of gravitational waves. We also study how to infer the SSE parameters using PTAs, and calculate the accuracy of parameter estimation.

Microarcsecond VLBI Pulsar Astrometry with PSRπ II. Parallax Distances for 57 Pulsars

Abstract We present the results of PSRπ, a large astrometric project targeting radio pulsars using the Very Long Baseline Array (VLBA). From our astrometric database of 60 pulsars, we have obtained parallax-based distance measurements for all but 3, with a parallax precision that is typically ∼45 μas and approaches 10 μas in the best cases. Our full sample doubles the number of radio pulsars with a reliable (≳5σ) model-independent distance constraint. Importantly, many of the newly measured pulsars are well outside the solar neighborhood, and so PSRπ brings a near-tenfold increase in the number of pulsars with a reliable model-independent distance at d &gt; 2 kpc. Our results show that both widely used Galactic electron density distribution models contain significant shortcomings, particularly at high Galactic latitudes. When comparing our results to pulsar timing, two of the four millisecond pulsars in our sample exhibit significant discrepancies in their proper motion estimates. With additional VLBI observations that extend our sample and improve the absolute positional accuracy of our reference sources, we will be able to additionally compare pulsar absolute reference positions between VLBI and timing, which will provide a much more sensitive test of the correctness of the solar system ephemerides used for pulsar timing. Finally, we use our large sample to estimate the typical accuracy attainable for differential VLBA astrometry of pulsars, showing that for sufficiently bright targets observed eight times over 18 months, a parallax uncertainty of 4 μas per arcminute of separation between the pulsar and calibrator can be expected.

Studying the Solar system with the International Pulsar Timing Array

Pulsar-timing analyses are sensitive to errors in the solar-system ephemerides (SSEs) that timing models utilise to estimate the location of the solar-system barycentre, the quasi-inertial reference frame to which all recorded pulse times-of-arrival are referred. Any error in the SSE will affect all pulsars, therefore pulsar timing arrays (PTAs) are a suitable tool to search for such errors and impose independent constraints on relevant physical parameters. We employ the first data release of the International Pulsar Timing Array to constrain the masses of the planet-moons systems and to search for possible unmodelled objects (UMOs) in the solar system. We employ ten SSEs from two independent research groups, derive and compare mass constraints of planetary systems, and derive the first PTA mass constraints on asteroid-belt objects. Constraints on planetary-system masses have been improved by factors of up to 20 from the previous relevant study using the same assumptions, with the mass of the Jovian system measured at 9.5479189(3)$\times10^{-4}$ $M_{\odot}$. The mass of the dwarf planet Ceres is measured at 4.7(4)$\times10^{-10}$ $M_{\odot}$. We also present the first sensitivity curves using real data that place generic limits on the masses of UMOs, which can also be used as upper limits on the mass of putative exotic objects. For example, upper limits on dark-matter clumps are comparable to published limits using independent methods. While the constraints on planetary masses derived with all employed SSEs are consistent, we note and discuss differences in the associated timing residuals and UMO sensitivity curves.

The NANOGrav 11 Year Data Set: Pulsar-timing Constraints on the Stochastic Gravitational-wave Background

Abstract We search for an isotropic stochastic gravitational-wave background (GWB) in the newly released 11 year data set from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). While we find no evidence for a GWB, we place constraints on a population of inspiraling supermassive black hole (SMBH) binaries, a network of decaying cosmic strings, and a primordial GWB. For the first time, we find that the GWB constraints are sensitive to the solar system ephemeris (SSE) model used and that SSE errors can mimic a GWB signal. We developed an approach that bridges systematic SSE differences, producing the first pulsar-timing array (PTA) constraints that are robust against SSE errors. We thus place a 95% upper limit on the GW-strain amplitude of A GWB &lt; 1.45 × 10−15 at a frequency of f = 1 yr−1 for a fiducial f −2/3 power-law spectrum and with interpulsar correlations modeled. This is a factor of ∼2 improvement over the NANOGrav nine-year limit calculated using the same procedure. Previous PTA upper limits on the GWB (as well as their astrophysical and cosmological interpretations) will need revision in light of SSE systematic errors. We use our constraints to characterize the combined influence on the GWB of the stellar mass density in galactic cores, the eccentricity of SMBH binaries, and SMBH–galactic-bulge scaling relationships. We constrain the cosmic-string tension using recent simulations, yielding an SSE-marginalized 95% upper limit of Gμ &lt; 5.3 × 10−11—a factor of ∼2 better than the published NANOGrav nine-year constraints. Our SSE-marginalized 95% upper limit on the energy density of a primordial GWB (for a radiation-dominated post-inflation universe) is ΩGWB(f) h 2 &lt; 3.4 × 10−10.

A dynamical approach in exploring the unknown mass in the Solar system using pulsar timing arrays

The error in the Solar system ephemeris will lead to dipolar correlations in the residuals of pulsar timing array for widely separated pulsars. In this paper, we utilize such correlated signals, and construct a Bayesian data-analysis framework to detect the unknown mass in the Solar system and to measure the orbital parameters. The algorithm is designed to calculate the waveform of the induced pulsar-timing residuals due to the unmodelled objects following the Keplerian orbits in the Solar system. The algorithm incorporates a Bayesian-analysis suit used to simultaneously analyse the pulsar-timing data of multiple pulsars to search for coherent waveforms, evaluate the detection significance of unknown objects, and to measure their parameters. When the object is not detectable, our algorithm can be used to place upper limits on the mass. The algorithm is verified using simulated data sets, and cross-checked with analytical calculations. We also investigate the capability of future pulsar-timing-array experiments in detecting the unknown objects. We expect that the future pulsar timing data can limit the unknown massive objects in the Solar system to be lighter than $10^{-11}$ to $10^{-12}$ $M_{\odot}$, or measure the mass of Jovian system to fractional precision of $10^{-8}$ to $10^{-9}$.