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is a pulsating white dwarf with a hydrogen atmosphere , a dav , also called a zz ceti star @xcite . these stars show multi - periodic non - radial @xmath5-mode pulsations that can be used to measure their internal properties and rate of evolution . @xcite found the star was variable , and @xcite studied its light curve , finding 6 simultaneous pulsations . the dominant mode has a period of 215  s , a fractional amplitude of 22 mma ( milli - modulation amplitude = 1/1.086 milli magnitude ) , and is stable in amplitude and phase . the other smaller pulsation modes vary in amplitude from night to night @xcite , suggesting the presence of unresolved components . because the davs appear to be normal stars except for their variability @xcite , i.e. , an evolutionary stage in the cooling of all white dwarfs , it is likely that the dav structural properties are representative of _ all _ da white dwarfs . @xcite and @xcite discovered 46 new zz cetis and more than doubled the number of known variables , using @xmath6 and @xmath7  g values derived from the optical spectra obtained by the sloan digital sky survey @xcite . while @xcite find the zz ceti instability strip to be pure , i.e. , contain no non - variable stars , @xcite find several stars inside the instability strip for which they could detect no variability above their detection threshold . it is necessary to obtain @xmath8 spectra of the non - variables within the strip and re - analyze their @xmath6 and @xmath7  g values , and also to obtain additional time series photometry on these stars , to ensure that they are not low amplitude variables , and re - study the purity of the instability strip . we report our continuing study of the star g  117b15a , also called ry lmi and wd  0921 + 354 , one of the hottest of the zz ceti stars . we expect that the rate of change of a pulsation period with time for @xmath5-mode pulsations in white dwarf stars to be directly related to its evolutionary timescale @xcite , allowing us to infer the age of a cool white dwarf since its formation . we have been observing the star since 1974 to measure the rate of period change with time ( @xmath9 ) for the largest amplitude periodicity . using all the data obtained from 1974 through 1999 , @xcite arrived at a determination of @xmath10 . g  117b15a was the first pulsating white dwarf to have its main pulsation mode index identified . the 215  s mode is an @xmath11 , as determined by comparing the ultraviolet pulsation amplitude ( measured with the hubble space telescope ) to the optical amplitude @xcite . @xcite confirm the @xmath12 measurement for the p=215  s pulsation , and show that the other large amplitude modes , at 271  s and 304  s , show chromatic amplitude changes that do not fit the theoretical models , using time - resolved spectra obtained at the keck telescope . @xcite , and @xcite derive @xmath6 near 12,400  k , while @xcite using a less efficient model for convection , derive @xmath6=11,630  k.    @xcite demonstrated that the observed variations in the light curve of g  117b15a are due to non - radial _ g_-mode pulsations . @xcite show the models predict the effect of radius change due to still ongoing contraction to be an order of magnitude smaller than the cooling effect on the rate of period change . g  117b15a is proving to be a useful laboratory for particle physics @xcite . @xcite calculated the limit on the axion mass compatible with the then observed upper limit to the cooling , showing @xmath13 and @xcite demonstrates axion cooling would be dominant over neutrino cooling for the lukewarm white dwarf stars for axion masses of this order . @xcite show that the @xmath14 upper limit published in @xcite limits the string mass scale @xmath15 for 6 dimensions , from the observed cooling rate and the emission of kaluza - klein gravitons , but the limit is negligible for higher dimensions . @xcite show the observed rates of period change can also be used to constrain the dynamical rate of change of the constant of gravity @xmath16 . @xcite used the mode identification and the observed periods of the 3 largest known pulsation modes to derive a hydrogen layer mass lower limit of @xmath17 , and a best estimate of @xmath18 , assuming @xmath19 for the 215  s mode , and 20:80 c / o core mass . the core composition is constrained mainly by the presence of the 304  s pulsation . @xcite show the seismological models with time - dependent element diffusion are only consistent with the spectroscopic data if the modes are the @xmath11 , k=2 , 3 , and 4 , and deduces @xmath20 , @xmath21 and @xmath22  k. their best model predicts : parallax @xmath23=15.89  mas , @xmath24  s / s , for the p=215  s , @xmath25  s / s , for the p=271  s , and @xmath26  s / s , for the p=304  s periodicities . @xcite reported on the observations from 1974 to 2000 . we report in this paper additional 19.3  h of time series photometry in 2001 , 30.6  h in 2002 , 24.4  h in 2003 , 4  h in 2004 , and 13.6  h in 2005 , most using the argos prime - focus ccd camera @xcite on the @xmath27  m otto struve telescope at mcdonald observatory . we observed the light through a bg40 filter to maximize the signal - to - noise ratio ( s / n ) because the pulsation amplitudes are small ( 2% ) , the star is faint ( v=15.52 , @xcite ) , and also because the non - radial _ g_-mode light variations have the same phase in all colors @xcite but the amplitudes decrease with wavelength . for example , a filter - less observation with argos gives an amplitude around 40% smaller for g  117b15a . we reduce and analyze the data in the manner described by @xcite , and @xcite . we bring all the data to the same fractional amplitude scale , and the times from utc to the uniform barycentric julian coordinated date ( tcb ) scale , using jpl de405 ephemeris @xcite to model earth s motion . we compute fourier transforms for each individual run , and verify that the main pulsation at 215  s dominates each data set and has an amplitude stable up to 15% , our uncertainty in amplitude due to the lack of accurate time- and color - dependent extinction determination . as the dominant pulsation mode at p=215  s has a stable frequency and amplitude since our first observations in 1974 , we can calculate the time of maximum for each new run and look for deviations . we fit our observed time of maximum light to the equation : @xmath28 where @xmath29 , @xmath30 , @xmath31 is the epoch of the time of maximum , i.e , the integer number of cycles after our first observation , @xmath32 is the time of maximum assumed , @xmath33 is the time of maximum that best fit the parabola , @xmath34 is the period that best fit the parabola , and @xmath35 is the period assumed at @xmath32 $ ] by non - linear least squares gives unreliable uncertainty estimates and the alias space in p and @xmath0 is extremely dense due to 31yr data set span @xcite . our non - linear least squares result is @xmath36s / s . ] . the times of maxima are calculated by a linear least - squares fit of the light curve of each night to a sum of the six detected frequencies . they are shown in table  [ tmax ] .    in figure [ figure1 ] , we show the o c timings after subtracting the correction to period and epoch , and our best fit curve through the data . the size of each point is proportional to its weight , i.e. , inversely proportional to the square of uncertainty in phase . the error bars plotted are @xmath37 . from our data through 2005 , we obtain a new value for the epoch of maximum , @xmath38 , a new value for the period , @xmath39 , and most importantly , a rate of period change of : @xmath40    we use linear least squares to make our fit , with each point weighted inversely proportional to the uncertainty in the time of maxima for each individual run squared . we quadratically add an additional 1s of uncertainty to the time of maxima for each night to account for external uncertainty caused perhaps by the beating of possible small amplitude pulsations @xcite or the small modulation seen in figure  [ figure1 ] . the amplitude , 1s , is chosen from the fourier transform of the ( o - c ) shown in figure  [ figure2 ] , and is in agreement with @xcite conclusion that the fourier analysis of the times of arrival ( toa ) gives unbiased information about the noise . such external uncertainty is also consistent with @xcite who show that the true uncertainties of the times of arrival of the milli - second pulsars are generally larger than the formal uncertainties , and that a quadratic term is added to them to fit the observations . while it is true that the period change timescale can be proportional to the cooling timescale , it is also possible that other phenomena with shorter timescales can affect @xmath9 . the cooling timescale is the longest possible one .    as a corollary , if the observed @xmath9 is low enough to be consistent with evolution , then other processes , such as perhaps a magnetic field or diffusion induced changes in the boundary layers , are not present at a level sufficient to affect @xmath9 . for the first time we also report on the search for the rates of period changes for the other relatively large amplitude modes of g  117b15a , @xmath41  s / s for the 270s periodicity , and @xmath42  s / s for the 304s periodicity . they are much larger than the one derived for the main pulsation , which has a region of period formation deeper in the core , but are in line with the measurements for the 274s modes in zz ceti @xcite . the models to date , without rotation , differential rotation , and magnetic fields , do not explain such values , or the chromatic amplitude changes reported by @xcite . @xcite show the 270s and 304s periodicities have significant amplitude changes , indicative of multiple components or resonances . @xcite discusses the influence of the proper motion of the star on the measured @xmath9 : @xmath43 where @xmath44 is the radial velocity of the star . assuming @xmath45 he derived @xmath46 \left(\mu[\," / yr]\right)^2 d[{\mathrm{pc}}]\ ] ] where @xmath47 is the effect of the proper motion on the rate of period change , @xmath34 is the pulsation period , @xmath48 is the proper motion and @xmath49 is the distance . the proper motion , @xmath50 , and the parallax , @xmath51 , are given by @xcite . but the parallax has a large uncertainty and does not agree with other estimates of the distance . @xcite measured the same proper motion for g  117-b15a and g  117-b15b : @xmath52  mas / yr , and @xmath53  mas / yr , including 5 usno - b epochs and sdss positions . considering @xmath54 @xcite , if we use @xcite estimate of @xmath55 derived from their optical spectra , we obtain a distance of @xmath56  pc , equivalent to a spectroscopic parallax of @xmath57 " . but g  117-b15a has international ultraviolet explorer ( iue ) and hubble space telescope ( hst ) faint object spectrograph ( fos ) flux calibrated spectra @xcite that can also be used to estimate the distance , if we use the evolutionary models of @xcite or @xcite to estimate the radius from @xmath6 and @xmath58 and fit the observed fluxes to a model atmosphere calculated by detlev koester , similar to that described in @xcite , but with an ml2/@xmath59=0.6 convection description consistent with @xcite and @xcite . the iue spectra re - calibrated according to the new spectroscopic image procession system ( newsips ) data reduction by nasa was published by @xcite . from the iue spectra we get a distance of @xmath60  pc . the hst spectra , shown in figure  [ hst ] , fits a distance of @xmath61  pc . table  [ dist ] lists all the available distances to g  117-b15a .    .g 117-b15a distance determinations [ cols= " < , > " , ]     the mass of an m3.5v should be around @xmath62 @xcite . with a separation around 13.4 arcsec @xmath63  au , corresponding to an orbital period around ( @xmath64 )  years , we get @xmath65 . the large uncertainty takes into account the possibility the orbit might be strongly elliptical . @xcite , in his section 4.3.1 , discusses common proper motion pairs which possibly do not form a physical binary . if g  117b15a and b form a real binary system , the contribution of an orbital reflex motion to the observed @xmath9 might account for one half of the observed @xmath0 .    the whole observed rate of period change _ could _ also be caused by a planet of jupiter s mass orbiting the white dwarf at a distance of 31  au , which corresponds to an orbital period of 223  yr , or a smaller planet on a closer orbit ( see figure  [ fergal ] ) . a planet with jupiter s mass any closer to the white dwarf would lead to a larger @xmath0 . @xcite show that such a planet would survive the post - main sequence mass loss . note that reflex motion produces periodic variations on the @xmath66 , which are distinguishable from parabolic variations after a significant portion of the orbit has been covered . as discussed by @xcite , any relative acceleration of the star with respect to the barycenter of the solar system will contribute to the observed @xmath0 . their equations ( 2.2 ) for the differential galactic orbits , decomposed in a planar contribution ( 2.12 ) , where the second term is the proper motion correction , and a perpendicular contribution ( 2.28 ) , applied to g  117-b15a , show the galactic contribution to be exactly the one calculated above for proper motion , i.e. , the other terms are negligible ( 2 to 3 orders of magnitude smaller ) . we have measured the rate of change of the main pulsation period for the @xmath67  k pulsating da white dwarf g  117b15a , the first zz ceti to have its evolutionary rate of change measured , confirming it is the most stable optical clock known , with a rate of change of 1  s in 8.9 million years and a precise laboratory for physics at high energy . it is important to notice that mode trapping , a resonance between the local pulsation wavelength with the thickness of one of the compositional layers , can reduce the rate of period change by up to a factor of 2 @xcite , but the changes in the trapping layers are still caused by cooling . it has taken a huge investment of telescope time to achieve such precision , but not only have we measured the cooling rate of this 400 million year old white dwarf @xcite , excluding the time the star took to reach the white dwarf phase , we have also demonstrated it does not harbor planetary bodies similar to jupiter in mass up to a distance around 30  au from the star or smaller planets with light travel time effects on the white dwarf larger than 1s . we claim that the 215s periodicity in g  117-b15a is the most stable optical clock known . @xcite and @xcite discuss projects to build optical atomic clocks based on single trapped ions , or several laser cooled neutron atoms , of strontrium or ytterbium , expected to reach an accuracy of @xmath68  s / s . considering their periods are @xmath69  s and @xmath70  s , even though they will be more accurate than g  117-b15a , they will be much less stable , as their timescales for period changes , @xmath71 , are 125  s and 3  h , compared to 2  gyr for g  117b15a . even the hulse & taylor s millisecond pulsar @xcite , has a timescale for period change of only 0.35  gyr @xcite , but the radio millisecond pulsar psr  j1713 + 0747 @xcite has @xmath72  s / s , and a timescale of 8  gyr , and psr  b1885 + 09=psr  j1857 + 0943 with @xmath73 s / s has a stability timescale of 9.5 gyr @xcite . together with these millisecond pulsars , g  117b15a may be used as the most stable known natural frequency and time - keeping standards [ e.g. @xcite ] . the white dwarf has the advantage of not having glitches and less severe general relativistic corrections . it is essential to note that the measured rate of period change is consistent with the cooling rate in our white dwarf models only for cores of c or c / o . with the refinement of the models , the observed rate of period change can be used to accurately measure the ratio of carbon to oxygen in the core of this normal white dwarf , and therefore help in constraining the @xmath74 cross section . 2442477.797089 & 32071 & 0.5 & 1.7 + 2442779.887934 & 153358 & 3.9 & 2.1 + 2442783.850624 & 154949 & 1.2 & 2.9 + 2442786.981458 & 156206 & 2.2 & 1.5 + 2443462.962774 & 427607 & 1.6 & 1.4 + 2443463.946592 & 428002 & 0.5 & 1.4 + 2443465.969049 & 428814 & 0.5 & 1.6 + 2443489.909755 & 438426 & 0.2 & 1.5 + 2443492.898616 & 439626 & 0.9 & 1.6 + 2443521.927837 & 451281 & 0.1 & 1.3 + 2443552.752879 & 463657 & 0.8 & 1.4 + 2443576.725940 & 473282 & -1.6 & 3.3 + 2443581.692438 & 475276 & 0.3 & 1.3 + 2443582.693698 & 475678 & -0.2 & 1.3 + 2443583.697469 & 476081 & 1.0 & 1.3 + 2443584.733602 & 476497 & 0.8 & 1.4 + 2443604.659292 & 484497 & 1.3 & 1.5 + 2443605.752703 & 484936 & 0.4 & 1.4 + 2443611.693050 & 487321 & 0.6 & 1.3 + 2443613.658222 & 488110 & 0.7 & 1.6 + 2443636.674971 & 497351 & 8.8 & 3.4 + 2443839.956765 & 578967 & 5.8 & 3.0 + 2443841.976708 & 579778 & 3.7 & 3.5 + 2443842.980413 & 580181 & -0.7 & 2.2 + 2443843.944332 & 580568 & 0.5 & 2.6 + 2443869.989703 & 591025 & 1.5 & 2.4 + 2443870.946182 & 591409 & 5.5 & 3.1 + 2443874.916339 & 593003 & 2.4 & 2.1 + 2443959.695117 & 627041 & 0.1 & 2.0 + 2443963.662836 & 628634 & 1.6 & 2.1 + 2443990.664641 & 639475 & 2.7 & 1.3 + 2444169.945954 & 711455 & 0.1 & 1.6 + 2444231.822666 & 736298 & -0.7 & 2.9 + 2444232.818992 & 736698 & 3.0 & 1.6 + 2444293.833896 & 761195 & 0.3 & 1.8 + 2444637.776174 & 899285 & 5.8 & 1.9 + 2444641.624287 & 900830 & 2.8 & 1.1 + 2444992.789531 & 1041820 & 0.1 & 1.6 + 2444994.689956 & 1042583 & 1.2 & 1.2 + 2444996.744801 & 1043408 & 2.0 & 1.3 + 2444997.723649 & 1043801 & 1.9 & 1.2 + 2445021.716661 & 1053434 & 1.7 & 1.4 + 2445703.860004 & 1327309 & 1.9 & 1.7 + 2445734.642701 & 1339668 & 2.4 & 1.2 + 2445735.643972 & 1340070 & 2.8 & 1.3 + 2446113.763716 & 1491882 & 2.9 & 1.2 + 2446443.775386 & 1624379 & 2.8 & 1.1 + 2446468.630178 & 1634358 & 2.1 & 1.3 + 2446473.718679 & 1636401 & 0.3 & 1.6 + 2446523.620086 & 1656436 & 2.2 & 1.6 + 2446524.613917 & 1656835 & 5.5 & 2.5 + 2446768.855451 & 1754896 & 2.9 & 1.4 + 2446794.935676 & 1765367 & 2.5 & 2.1 + 2446796.928219 & 1766167 & 0.3 & 1.6 + 2446797.924535 & 1766567 & 3.1 & 1.3 + 2446798.903378 & 1766960 & 2.6 & 1.8 + 2446823.663537 & 1776901 & 3.1 & 1.9 + 2446825.651132 & 1777699 & 3.7 & 1.5 + 2447231.328096 & 1940575 & 3.7 & 1.9 + 2447231.612054 & 1940689 & 5.1 & 3.5 + 2447232.396626 & 1941004 & 5.0 & 1.6 + 2447232.623291 & 1941095 & 5.9 & 2.9 + 2447233.343090 & 1941384 & 4.5 & 1.3 + 2447233.634506 & 1941501 & 4.7 & 2.3 + 2447234.319475 & 1941776 & 6.8 & 3.2 + 2447235.313250 & 1942175 & 5.2 & 1.4 + 2447235.607168 & 1942293 & 6.4 & 2.1 + 2447236.610922 & 1942696 & 6.2 & 1.6 + 2447589.375198 & 2084328 & 3.2 & 1.4 + 2447594.331735 & 2086318 & 5.2 & 1.6 + 2447595.323018 & 2086716 & 3.5 & 2.0 + 2447596.311907 & 2087113 & 10.1 & 2.3 + 2447597.315602 & 2087516 & 4.8 & 1.7 + 2447598.319339 & 2087919 & 3.1 & 3.1 + 2447499.072036 & 2048072 & 6.5 & 3.2 + 2447532.768799 & 2061601 & 1.3 & 1.4 + 2447853.846325 & 2190511 & 4.3 & 2.1 + 2447856.832697 & 2191710 & 5.2 & 1.9 + 2447918.644630 & 2216527 & 2.6 & 3.1 + 2447920.619811 & 2217320 & 6.7 & 3.3 + 2447952.622834 & 2230169 & -3.3 & 2.9 + 2447972.620899 & 2238198 & 9.6 & 6.1 + 2447973.709340 & 2238635 & 9.7 & 2.6 + 2447973.741682 & 2238648 & 6.5 & 1.4 + 2447978.770467 & 2240667 & 10.0 & 2.1 + 2447979.781717 & 2241073 & 11.8 & 3.1 + 2447980.319627 & 2241289 & 4.6 & 3.5 + 2447977.403038 & 2240118 & 7.5 & 2.3 + 2447978.327055 & 2240489 & 4.3 & 3.3 + 2447979.358189 & 2240903 & 2.6 & 3.4 + 2447979.358145 & 2240903 & -1.2 & 4.9 + 2447978.601069 & 2240599 & 7.4 & 2.5 + 2447980.621017 & 2241410 & 5.8 & 3.4 + 2447980.782929 & 2241475 & 7.2 & 2.3 + 2447981.325918 & 2241693 & 8.4 & 1.4 + 2447981.592393 & 2241800 & 5.7 & 1.4 + 2447981.779185 & 2241875 & 4.8 & 1.1 + 2447982.329663 & 2242096 & 7.4 & 1.8 + 2447982.743093 & 2242262 & 5.0 & 1.2 + 2447983.734400 & 2242660 & 5.4 & 1.2 + 2447979.281057 & 2240872 & 9.5 & 2.9 + 2447980.224899 & 2241251 & -2.4 & 2.9 + 2447984.735678 & 2243062 & 6.5 & 1.1 + 2448245.724666 & 2347847 & -3.3 & 5.1 + 2448267.799932 & 2356710 & 5.2 & 2.3 + 2448324.627972 & 2379526 & 4.3 & 1.2 + 2448325.708938 & 2379960 & 4.1 & 1.3 + 2448328.593208 & 2381118 & 6.4 & 1.6 + 2448331.661735 & 2382350 & 4.0 & 1.2 + 2448238.571479 & 2344975 & 8.3 & 2.2 + 2448622.833258 & 2499253 & 3.3 & 1.8 + 2448680.642683 & 2522463 & 6.3 & 1.2 + 2448687.614155 & 2525262 & 4.0 & 1.2 + 2448688.597979 & 2525657 & 3.4 & 1.2 + 2449062.660365 & 2675840 & 4.2 & 1.6 + 2449063.609354 & 2676221 & 6.7 & 1.9 + 2449066.615640 & 2677428 & 6.5 & 1.4 + 2449066.371558 & 2677330 & 7.2 & 2.0 + 2449066.326737 & 2677312 & 8.2 & 2.6 + 2449069.342967 & 2678523 & 6.4 & 1.7 + 2449298.239287 & 2770423 & 8.5 & 4.1 + 2449298.304041 & 2770449 & 8.2 & 4.1 + 2449294.214264 & 2768807 & 5.5 & 4.1 + 2449294.293897 & 2768839 & -0.5 & 4.1 + 2449295.439583 & 2769299 & -4.0 & 6.1 + 2449295.494387 & 2769321 & -3.3 & 7.1 + 2449036.809260 & 2665461 & 2.4 & 2.2 + 2449038.677300 & 2666211 & 3.1 & 2.2 + 2449040.687310 & 2667018 & 3.6 & 4.1 + 2449041.616360 & 2667391 & 4.9 & 4.1 + 2449799.723888 & 2971765 & 5.6 & 1.3 + 2450427.920960 & 3223981 & 8.2 & 3.8 + 2450429.973242 & 3224805 & 2.7 & 2.4 + 2450430.914779 & 3225183 & 6.9 & 2.5 + 2450431.843821 & 3225556 & 7.5 & 1.5 + 2450434.912392 & 3226788 & 8.8 & 2.0 + 2450436.929828 & 3227598 & 5.4 & 1.7 + 2450483.633189 & 3246349 & 9.6 & 1.8 + 2451249.5989069 & 3553878&10.1&1.3 + 2451249.7632895 & 3553944&9.7 & 1.7 + 2451250.6126098 & 3554285&8.7&2.0 + 2451526.8772586 & 3665203 & 10.4 & 1.2 + 2451528.8523866 & 3665996&10.0&1.5 + 2451528.9196061 & 3666023&7.4&1.4 + 2451528.9868422 & 3666050&6.3&1.9 + 2451529.8585943 & 3666400 & 6.6&2.0 + 2451530.9097492 & 3666822 & 13.4&2.41 + 2451960.8561629 & 3839442 & 10.1&1.62 + 2451962.7864775 & 3840217 & 11.3&1.48 + 2451967.6806926 & 3842182 & 8.1 & 1.93 + 2451988.7919772 & 3850658 & 10.5 & 2.00 + 2451990.7845255 & 3851458 & 8.8 & 1.59 + 2452037.6472583 & 3870273 & 10.1 & 3.39 + 2452045.6399770 & 3873482 & 12.5&1.85 + 2452225.9050927 & 3945857 & 7.6 & 1.34 + 2452225.9598927 & 3945879 & 8.0 & 0.65 + 2452263.8834810 & 3961105 & 10.6&0.58 + 2452316.6442205 & 3980721 & 13.1&1.0 + 2452317.8995164 & 3982288 & 12.2&0.67 + 2452319.7999417 & 3982691 & 12.0&0.79 + 2452317.6479750 & 3982792 & 10.3&0.95 + 2452321.8348344 & 3983555 & 11.4&1.23 + 2452322.7265266 & 3984372 & 9.9 & 3.03 + 2452312.7412881 & 3984730 & 11.4&3.5 + 2452373.6840808 & 4089983 & 9.9&1.1 + 2452373.6839702 & 4090425 & 12.6&1.2 + 2452373.7140655 & 4090791 & 10.6&0.68 + 2452375.6392709 & 4122465 & 12.8&1.0 + 2452374.7700070 & 4124076 & 12.5&1.23 + 2452581.9494464 & 4134940 & 12.0 & 1.83 + 2452583.9095168 & 4137288 & 11.5 & 1.74 + 2452584.8812628 & 4144503 & 13.8 & 0.92 + 2452585.9821875 & 4148953 & 12.0 & 1.16 + 2452586.8937641 & 4146104 & 12.3 & 0.7 + 2452665.7845581 & 4255806 & 13.7 & 0.72 + 2452669.7970851 & 4256260 & 10.6 & 1.52 + 2452696.8561592 & 4257768 & 12.8 & 0.96 + 2452724.6624548 & 4267394 & 10.5 & 0.81 + 2453381.7442572 & 4409917 & 11.5 & 1.41 + 2453439.7653860 & 4433212 & 13.9 & 1.0 + 2453446.6073830 & 4435959 & 15.2 & 0.71 + 2453473.6191370 & 4446804 & 15.1 & 0.75 +
we report our measurement of the rate of change of period with time ( @xmath0 ) for the 215s periodicity in the pulsating white dwarf g  117b15a , the most stable optical clock known . after 31 years of observations 
 , we have finally obtained a 4@xmath1 measurement @xmath2 . 
 taking into account the proper - motion effect of @xmath3 , we obtain a rate of change of period with time of @xmath4 . 
 this value is consistent with the cooling rate in our white dwarf models only for cores of c or c / o . with the refinement of the models , 
 the observed rate of period change can be used to accurately measure the ratio of c / o in the core of the white dwarf .