Understanding the Predication Mechanism of Deep Learning through Error Propagation among Parameters in Strong Lensing Case

The error propagation among estimated parameters reflects the correlation among the parameters.We study the capability of machine learning of"learning"the correlation of estimated parameters.We show that machine learning can recover the relation between the uncertainties of different parameters,especially,as predicted by the error propagation formula.Gravitational lensing can be used to probe both astrophysics and cosmology.As a practical application,we show that the machine learning is able to intelligently find the error propagation among the gravitational lens parameters(effective lens mass ML and Einstein radiusθ_(E))in accordance with the theoretical formula for the singular isothermal ellipse(SIE)lens model.The relation of errors of lens mass and Einstein radius,(e.g.,the ratio of standard deviations F=σ_(ML)/σ_(θ_(E)))predicted by the deep convolution neural network are consistent with the error propagation formula of the SIE lens model.As a proof-of-principle test,a toy model of linear relation with Gaussian noise is presented.We found that the predictions obtained by machine learning indeed indicate the information about the law of error propagation and the distribution of noise.Error propagation plays a crucial role in identifying the physical relation among parameters,rather than a coincidence relation,therefore we anticipate our case study on the error propagation of machine learning predictions could extend to other physical systems on searching the correlation among parameters.

Correction Factors of the Measurement Errors of the LAMOST-LRS Stellar Parameters

We aim to investigate the propriety of stellar parameter errors of the official data release of the LAMOST lowresolution spectroscopy(LRS)survey.We diagnose the errors of radial velocity(RV),atmospheric parameters([Fe/H],Teff,logg)andα-enhancement([α/M])for the latest data release version of DR7,including 6,079,235effective spectra of 4,546,803 stars.Based on the duplicate observational sample and comparing the deviation of multiple measurements to their given errors,we find that,in general,the error of[α/M]is largely underestimated,and the error of RV is slightly overestimated.We define a correction factor k to quantify these misestimations and correct the errors to be expressed as proper internal uncertainties.Using this self-calibration technique,we find that the k-factors significantly vary with the stellar spectral types and the spectral signal-to-noise ratio(S/N).Particularly,we reveal a strange but evident trend between k-factors and error themselves for all five stellar parameters.Larger errors tend to have smaller k-factor values,i.e.,they were more overestimated.After the correction,we recreate and quantify the tight correlations between S/N and errors,for all five parameters,while these correlations have dependence on spectral types.It also suggests that the parameter errors from each spectrum should be corrected individually.Finally,we provide the error correction factors of each derived parameter of each spectrum for the entire LAMOST-LRS DR7 and plan to update them for the later data releases.

Relations between SMBH Parameters and Jet Generation and Efficiency in Blazars

We analyzed the relationship between several basic parameters describing supermassive black holes such as jet power,black hole spin,accretion disk magnetic field,black hole mass,etc.We found that there is a general correlation between these parameters,such as jet power is significantly positively correlated with black hole spin,while black hole mass is significantly negatively correlated with black hole spin.To apprehend these relationships,we consider the Blandford–Znajek model to be superior to the Blandford–Payne model.It is also found that the intrinsic gamma luminosity of the FSRQs has a positive correlation with the accretion disk magnetic field,while the intrinsic gamma luminosity of the BL Lacs has a negative correlation with the accretion disk magnetic field.A feedback effect may exist between accretion disk accretion rate and magnetic field,which may be the key to the evolution between BL Lacs and FSRQs.There is no significant difference in the jet power and jet generation efficiency of FSRQs and BL Lacs,which suggests that the jets are generated by the same mechanism.The contribution rate of accretion rate to jet generation efficiency is high,while the contribution rate of accretion rate to jet power is very low.

Two-dimensional Parameter Relationships for W UMa-type Systems Revisited

Reviewing the empirical and theoretical parameter relationships between various parameters is a good way to understand more about contact binary systems.In this investigation,two-dimensional(2D)relationships for P–MV(system),P–L1,2,M1,2–L1,2,and q–Lratiowere revisited.The sample used is related to 118 contact binary systems with an orbital period shorter than 0.6 days whose absolute parameters were estimated based on the Gaia Data Release 3 parallax.We reviewed previous studies on 2D relationships and updated six parameter relationships.Therefore,Markov chain Monte Carlo and Machine Learning methods were used,and the outcomes were compared.We selected 22 contact binary systems from eight previous studies for comparison,which had light curve solutions using spectroscopic data.The results show that the systems are in good agreement with the results of this study.

Physical and geometrical parameters of CVBS Ⅺ: Cou 1511(HIP 12552)

Model atmospheres of the close visual binary star Cou 1511(HIP 12552) are constructed using grids of Kuruz's blanketed models to build the individual synthetic spectral energy distributions(SEDs) for both components. These synthetic SEDs are combined together for the entire system and compared with the observational one following Al-Wardat's complex method for analyzing close visual binary stars. The entire observational SED of the system is used as a reference for comparison between the synthetic SED and the observed one. The parameters of both components are derived as:T_(eff)~a= 6180 ± 50 K,T_(eff)~b=5865 ± 70 K, log g_a = 4.35 ± 0.12, log g_b = 4.45 ± 0.14, R_a = 1.262 ± 0.08 R_⊙, R_b = 1.006 ± 0.07 R_⊙,L_a = 2.09 ± 0.10 L_⊙and L_b = 1.08 ± 0.12 L_⊙, with spectral types F8 V and G1.5V for components(a,b) respectively and age of 3.0 ± 0.9 Gyr. A modified orbit of the system is built and the masses of the two components are calculated as M_a = 1.17 ± 0.11 M_⊙and M_b = 1.06 ± 0.10 M_⊙.

Absolute parameters of the three totally-eclipsing W UMa stars NSVS 2443858,NSVS 780649 and V1098 Her

We present the first photometric analysis of three totally-eclipsing W UMa binaries,NS VS2443858,NSVS 780649 and V1098 Her.The absolute astrophysical parameters of the stellar components were determined by means of Gaia distances and light curve solutions.The results show that:(ⅰ)Two of the systems,NSVS 2443858 and V1098 Her,are of A subtype while the obtained temperature of the secondary component of NSVS 780649 indicates that it is a W-subtype system;(ⅱ)The estimated mass ratios approach the lower limit of the mass ratio assumed by researchers in recent years so our targets could be classified as extreme mass ratio binary(EMRB)systems;(ⅲ)All the systems have deep contact configurations,so they also are deep low mass ratio(DLMR)systems;(ⅳ)The components of our systems are stars of F and G spectral type and undergo total eclipses;(ⅴ)The sum 0.871 M⊙of the component masses of NSVS 780649 is below the mass limit of 1.0-1.2 M⊙assumed for the known contact binary stars.

Irregular Spatial Distributions of Spectral Line Parameters in the Middle Solar Chromosphere Revealed from Analysis of Solar Flash MgⅠb_(2)Spectra

This paper presents the analytical results of solar MgⅠb_(2)flash spectra,obtained by the prototype Fiber Arrayed Solar Optic Telescope in process of the 2013 Gabon total solar eclipse.The analysis reveals irregular distributions of the spectral line parameters like ratio of line source function to continuum oneβ,ratio of line emissivity to continuum emissivityζ,ratio of the continuum opacity to the line opacity r_(0),line center optical depthτ_(0)、,the line widthΔλ_(D),and the line-of-sight velocity v_(los),while the approximately spherical symmetry can be found in the maps of integrated line intensity and continuum intensity.These irregular distributions originate from those of line profile features like the maximum intensity,the line width and line center wavelength.It is also found from the recovered line center optical depthτ_(0) that in the middle chromosphere,the optical depth is not small due to nonignorable absorption and the long light path along the line-of-sight.Finally,we show that the excessive broadening of spectral lines can be due to co-existence of multiple radiative sources with different line-of-sight velocities unresolved in one detector pixel.

Absolute physical parameters of three poorly studied detached eclipsing binaries

The photometric and spectroscopic data for three double-lined detached eclipsing binaries were collected from the photometric and spectral surveys.The light and radial velocity curves of each binary system were simultaneously analyzed using Wilson-Devinney(WD)code,and the absolute physical and orbital parameters of these binaries were derived.The masses of both components of ASASSN-V J063123.82+192341.9 were found to be M1=1.088±0.016 and M20.883±0.016 M⊙;those of ASAS J011416+0426.4 were determined to be M1=0.934±0.046 and M2=0.754±0.043 M⊙;and those of MW Aur were derived to be M1=2.052±0.196 and M21.939±0.193 M⊙.Finally,the evolutionary status of these detached binaries was discussed based on their absolute parameters and the theoretical stellar models.

Physical and geometrical parameters of CVBS XIV:the two nearby systems HIP 19206 and HIP 84425

Data release 2(DR2) from the Gaia mission was of great help in precise determination of fundamental parameters of Close Visual Binary and Multiple Systems(CVBMSs), especially masses of their components, which are crucial parameters in understanding formation and evolution of stars and galaxies. This article presents the complete set of fundamental parameters for two nearby close visual binary systems(CVBSs), which are HIP 19206 and HIP 84425. We utilised a combination of two methods;the first one is Tokovinin's dynamical method to solve the orbit of the system and to estimate orbital elements and the dynamical mass sum, and the second one is Al-Wardat's method for analysing CVBMSs to estimate the physical parameters of the individual components. The latest method employs grids of Kurucz lineblanketed plane parallel model atmospheres to build synthetic Spectral Energy Distributions(SEDs) of the individual components. Trigonometric parallax measurements given by Gaia DR2 and Hipparcos catalogues are used to analyse the two systems. The difference in these measurements yielded slight discrepancies in the fundamental parameters of the individual components, especially masses. So, a new dynamical parallax is suggested in this work based on the most convenient mass sum as given by each of the two methods. The new dynamical parallax for the system HIP 19205 of 22.97±0.95 mas coincides well with the trigonometric one reported recently(in December 2020) by Gaia EDR3 of 22.3689 ± 0.4056 mas. The positions of the components of the two systems on the evolutionary tracks and isochrones are plotted, which suggest that all components are solar-type main sequence stars. Their most probable formation and evolution scenarios are also discussed.

Estimation of Stellar Atmospheric Parameters from LAMOST DR8 Low-resolution Spectra with 20 ≤ S/N < 30

The accuracy of the estimated stellar atmospheric parameter evidently decreases with the decreasing of spectral signal-to-noise ratio(S/N)and there are a huge amount of this kind observations,especially in case of S/N<30.Therefore,it is helpful to improve the parameter estimation performance for these spectra and this work studied the(T_(eff),log g,[Fe/H])estimation problem for LAMOST DR8 low-resolution spectra with 20≤S/N<30.We proposed a data-driven method based on machine learning techniques.First,this scheme detected stellar atmospheric parameter-sensitive features from spectra by the Least Absolute Shrinkage and Selection Operator(LASSO),rejected ineffective data components and irrelevant data.Second,a Multi-layer Perceptron(MLP)method was used to estimate stellar atmospheric parameters from the LASSO features.Finally,the performance of the LASSO-MLP was evaluated by computing and analyzing the consistency between its estimation and the reference from the Apache Point Observatory Galactic Evolution Experiment high-resolution spectra.Experiments show that the Mean Absolute Errors of T_(eff),log g,[Fe/H]are reduced from the LASP(137.6 K,0.195,0.091 dex)to LASSO-MLP(84.32 K,0.137,0.063 dex),which indicate evident improvements on stellar atmospheric parameter estimation.In addition,this work estimated the stellar atmospheric parameters for 1,162,760 lowresolution spectra with 20≤S/N<30 from LAMOST DR8 using LASSO-MLP,and released the estimation catalog,learned model,experimental code,trained model,training data and test data for scientific exploration and algorithm study.

Influence of Solar Activity on Precise Orbit Prediction of LEO Satellites

The perturbations of low earth orbit(LEO)satellites operating in the orbit of 300~2000 km are complicated.In particular,the atmospheric drag force and solar radiation pressure force change rapidly over a short period of time due to solar activities.Using spaceborne global positioning system(GPS)data of the CHAMP,GRACE and SWARM satellites from 2002 to 2020,this paper studies in depth the influence of solar activity on LEO satellites’precise orbit prediction by performing a series of orbit prediction experiments.The quality of GPS data is more susceptible to being influenced by solar activity during years when this activity is high and the changes in dynamic parameters are consistent with those of solar activity.The effects of solar activity on LEO orbit prediction accuracy are analyzed by comparing the predicted orbits with the precise ones.During years of high solar activity,the average root-mean-squares prediction errors at 10,20,and 30 minutes are 0.15,0.20,and 0.26 m,respectively,which are larger than the corresponding values in low-solar-activity years by 59%,63%,and 68%,respectively.These results demonstrate that solar activity has a great influence on the orbit prediction accuracy,especially during high-solar-activity years.We should strengthen the real-time monitoring of solar activity and geomagnetic activity,and formulate corresponding orbit prediction strategies for the active solar period.

An Update of the Catalog of Radial Velocity Standard Stars from the APOGEE DR17

We present an updated catalog of 46,753 radial velocity(RV)standard stars selected from the APOGEE DR17.These stars cover the Northern and Southern Hemispheres almost evenly,with 62%being red giants and 38%being main sequence stars.These RV standard stars are stable on a baseline longer than 200 days(with 54%longer than one year and 10%longer than five years)with a median stability better than 215 m s~(-1).The average number of observations of those stars is 5 and each observation is required to have signal-to-noise ratio greater than50 and RV measurement error smaller than 500 m s~(-1).Based on the new APOGEE RV standard star catalog,we have checked the RV zero-points(RVZPs)for current large-scale stellar spectroscopic surveys including RAVE,LAMOST,GALAH and Gaia.By careful analysis,we estimate their mean RVZP to be+0.149 kms~(-1),+4.574 km s~(-1)(for LRS),-0.031 km s~(-1)and+0.014 kms~(-1),respectively,for the four surveys.In the RAVE,LAMOST(for MRS),GALAH and Gaia surveys,RVZP exhibits a systematic trend with stellar parameters(mainly[Fe/H],T_(eff),log g,G_(BP)-G_(RP)and G_(RVS)).The corrections to those small but clear RVZPs are of vital importance for these massive spectroscopic surveys in various studies that require extremely high RV accuracies.

Hybrid Optimization Algorithm in the Photometric Inversion of Periods for Asteroids

Asteroids,as the primitive building blocks for the formation of our solar system,could reveal its evolution mechanism,and have attracted more and more attention from the public and professional institutions in recent years.Their physical properties,such as rotational period,spin axis and overall shape,can be inverted from ground-and space-based photometric observations.Since the inversion process is very time-consuming,this paper combines the genetic algorithm with the Levenberg–Marquardt(LM) algorithm,and presents a hybrid optimization algorithm based on a Cellinoid shape model for the inversion of rotational periods,which greatly improves the inversion efficiency.The proposed hybrid algorithm is applied to the synthetic lightcurves generated for an assumed Cellinoid shape model and the inverted rotational period results are consistent with the preset ones with a reduced search time,compared with the LM algorithm.Finally,multiple numerical experiments on the periods are performed on lightcurves and sparse observations of real asteroids to confirm that the proposed method can perform well in improving computational efficiency.

An Independent Degree-eight Mars Gravity Field Model and the Expected Results from the Tianwen-1 Mission

Tianwen-1 is China's first independent interplanetary exploration mission,targeting Mars,and includes orbiting,landing,and rover phases.Similar to previous Mars missions,the Tianwen-1 orbiter was designed for polar orbits during the scientific mission period but has an exceptional eccentricity of approximately 0.59.We provide the first independent eight-degree Martian gravity field model in this paper,which was developed exclusively by a team working in China with our independent software as well,based on about two months of radiometric Doppler and range data from only the Tianwen-1 mission.This model is independent from the models created by the groups at NASA Jet Propulsion Laboratory and Goddard Space Flight Center in the United States,as well as the Centre National d'Etudes Spatiales in France.Furthermore,in order to optimize the engineering and scientific benefits,we proposed a number of potential orbits for the extended Tianwen-1 mission.In order to solve a higher-degree independent Mars gravity field model,the viability of modifying the perigee height was investigated,with the priority considerations of fuel savings and implementation hazards being controlled.

Transverse Velocity Field Measurements in High-resolution Solar Images Based on Deep Learning

To address the problem of the low accuracy of transverse velocity field measurements for small targets in highresolution solar images,we proposed a novel velocity field measurement method for high-resolution solar images based on PWCNet.This method transforms the transverse velocity field measurements into an optical flow field prediction problem.We evaluated the performance of the proposed method using the Hαand TiO data sets obtained from New Vacuum Solar Telescope observations.The experimental results show that our method effectively predicts the optical flow of small targets in images compared with several typical machine-and deeplearning methods.On the Hαdata set,the proposed method improves the image structure similarity from 0.9182 to0.9587 and reduces the mean of residuals from 24.9931 to 15.2818;on the TiO data set,the proposed method improves the image structure similarity from 0.9289 to 0.9628 and reduces the mean of residuals from 25.9908 to17.0194.The optical flow predicted using the proposed method can provide accurate data for the atmospheric motion information of solar images.The code implementing the proposed method is available on https://github.com/lygmsy123/transverse-velocity-field-measurement.

First Detailed Photometric Investigation on the Nature of Contact Binary System AA Cet

The TESS light curve(LC)of the marginal contact binary A A Cet was analyzed simultaneously with the radial velocity and the orbital period(OP)change of the system was investigated.The physical parameters of the system were obtained by analyzing the LC of A A Cet with the Wilson-Devinney method,and the absolute parameters of the components were calculated using the results obtained.For the components of AA Cet,the masses and radii were calculated as M_(1)=1.39±0.04 M_⊙,M_(2)=0.48±0.02 M_⊙,and R_(1)=1.64±0.03 R_⊙,R_(2)=1.01±0.04 R_⊙,respectively.AA Cet is a marginal contact binary with a temperature difference of 1305 K between its components.A total of 14 eclipse times were obtained from the TESS data and used in the OP analysis together with those collected from the literature.It has been observed that the change in the OP of AA Cet is in the form of a decreasing parabola.Conservative mass transfer between the components has been interpreted as the reason for this change.The OP decrease amount of A A Cet was obtained as dP/dt=0.0062±0.0006 s yr^(-1).and the reason for this decrease was attributed to a 3.3(9)×10^(-8)M_⊙.mass transfer per year from the more massive component to the less massive one.The age of A A Cet has been estimated as 7 Gyr,as the age of contact systems helps us to understand their evolution.

On the Relation Between Coronal Green Line Brightness and Magnetic Fields Intensity

Two-dimensional(2 D) solar coronal magnetogram is difficult to be measured directly until now.From the previous knowledge,a general relation has been noticed that the brighter green-line brightness for corona,the higher coronal magnetic field intensity may correspond to.To try to further reveal the relationship between coronal green line brightness and magnetic field intensity,we use the 2 D coronal images observed by Yunnan Observatories Greenline Imaging System(YOGIS) of the 10 cm Lijiang coronagraph and the coronal magnetic field maps calculated from the current-free extrapolations with the photospheric magnetograms taken by Helioseismic and Magnetic Imager(HMI) on board the Solar Dynamics Observatory(SDO) spacecraft.In our analysis,we identified the coronal loop structures and construct two-dimensional maps of the corresponding magnetic field intensity in the plane of the sky(POS) above the limb.We derive the correlation coefficients between the coronal brightness and the magnetic field intensity for different heights of coronal layers.We further use a linear combination of a Gaussian and a quadratic profile to fit the correlation coefficients distribution,finding a largest correlation coefficient of 0.82 near 1.1 R(solar radii) where is almost the top of the closed loop system.For the small closed loop system identified,the correlation coefficient distributions crossing and covering the loop are calculated.We also investigate the correlation with extended heliocentric latitude zones and long period of one whole Carrington Rotation,finding again that the maximum correlation coefficient occurs at the same height.It is the first time for us to find that the correlation coefficients are high(all are larger than 0.8) at the loop-tops and showing poor correlation coefficients with some fluctuations near the feet of the coronal loops.Our findings indicate that,for the heating of the low-latitude closed loops,both DC(dissipation of currents) and AC(dissipation of Alfvén and magnetosonic waves) mechanisms should act simultaneously on the whole closed loop system while the DC mechanisms dominate in the loop-top regions.Therefore,in the distributions of the correlation coefficients with different heights of coronal layers,for both large-and small-scale latitude ranges,the coefficients can reach their maximum values at the same coronal height of 1.1 R,which may indicate the particular importance of the height of closed loops for studying the coupling of the local emission mechanism and the coronal magnetic fields,which maybe helpful for studying the origin of the low-speed solar wind.

Luminosity of a radio pulsar and its new emission death line

We investigated the pulsar radio luminosity(L),emission efficiency(ratio of radio luminosity to its spin-down powerE),and death line in the magnetic field(B)versus spin period(P)diagram.We found that the dependence of pulsar radio luminosity on its spin-down power(L-E)is very weak,shown as L^E0.06,which deduces an equivalent inverse correlation between emission efficiency and spin-down power asξ~E-0.94.Furthermore,we examined the distributions of radio luminosity of millisecond and normal pulsars and found that for the similar spin-down powers,the radio luminosity of millisecond pulsars is about one order of magnitude lower than that of the normal pulsars.The analysis of pulsar radio flux suggests that these correlations are not due to a selective effect but are intrinsic to the pulsar radio emission physics.Their radio radiations may be dominated by the different radiation mechanisms.The cutoff phenomenon of currently observed radio pulsars in B-P diagram is usually referred as the"pulsar death line",which corresponds toE≈1030 erg s-1 and is obtained by the cut-off voltage of electron acceleration gap in the polar cap model of pulsar proposed by Ruderman and Sutherland.Observationally,this death line can be inferred by the actual observed pulsar flux S≥1 mJy and 1 kpc distance,together with the maximum radio emission efficiency of 1%.However,the observation data show that the 37 pulsars pass over the death line,including the recently observed two pulsars with long periods of 23.5 s and 12.1 s,which violate the prediction of the polar cap model.At present,the actual observed pulsar flux can reach0.01 mJy by FAST telescope.This will arise the observational limit of spin-down power of pulsars as low as E˙≈1028 erg s-1.This means that the new death line is downward shifted two orders of magnitude,which might be favorably referred as the"observational limit–line".Accordingly,the pulsar theoretical model for the cut-off voltage of gap should be heavily modified.

A star-based method for precise wavelength calibration of the Chinese Space Station Telescope(CSST) slitless spectroscopic survey

The Chinese Space Station Telescope(CSST)spectroscopic survey aims to deliver high-quality low-resolution(R>200)slitless spectra for hundreds of millions of targets down to a limiting magnitude of about 21 mag,distributed within a large survey area(17500 deg2)and covering a wide wavelength range(255-1000 nm by three bands GU,GV,and GI).As slitless spectroscopy precludes the usage of wavelength calibration lamps,wavelength calibration is one of the most challenging issues in the reduction of slitless spectra,yet it plays a key role in measuring precise radial velocities of stars and redshifts of galaxies.In this work,we propose a star-based method that can monitor and correct for possible errors in the CSST wavelength calibration using normal scientific observations,taking advantage of the facts that(ⅰ)there are about ten million stars with reliable radial velocities now available thanks to spectroscopic surveys like LAMOST,(ⅱ)the large field of view of CSST enables efficient observations of such stars in a short period of time,and(ⅲ)radial velocities of such stars can be reliably measured using only a narrow segment of CSST spectra.We demonstrate that it is possible to achieve a wavelength calibration precision of a few km s^(-1) for the GU band,and about 10 to 20 kms^(-1) for the GV and GI bands,with only a few hundred velocity standard stars.Implementations of the method to other surveys are also discussed.

Seismic diagnostics of solar-like oscillating stars

High precision and long-lasting Kepler data enabled us to estimate stellar properties with asteroseismology as an accurate tool. We performed asteroseismic analysis on six solar-like stars observed by the Kepler mission: KIC 6064910, KIC 6766513, KIC 7107778, KIC 10079226, KIC 10147635 and KIC 12069127. The extraction of seismic information includes two parts. First, we obtained two global asteroseismic parameters, mean large separation ?_ν and frequency of maximum power ν_(max),with autocorrelation function and collapsed autocorrelation function. Second, we extracted individual oscillation modes ν_(nl) with low-l degree using a least-squares fit. Stellar grid models were built with Yale Rotating Stellar Evolution Code(YREC) to analyze stellar properties. They covered the range of M = 0.8 ~ 1.8 M_⊙with a step of 0.02 M_⊙ and [Fe/H] =-0.3 ~ 0.4 dex with a step of 0.1 dex.We used a Bayesian approach to estimate stellar fundamental parameters of the six stars, under the constraints of asteroseismic parameters(?_ν, ν_(max)) and non-asteroseismic parameters(Teff, [Fe/H]). We discover that the six targets include five sub-giant stars with 1.2 ~1.5 M_⊙ and one main-sequence star with 1.08 M_⊙, and with ages in the range of 3 ~5 Gyr.