Observing Strategies for the Detection of Jupiter Analogs

From the main AAPS sample of $\sim$250 stars, we wish to define a subsample of stars which have both low intrinsic stellar activity and a sufficient observational baseline to be useful for the detection of Jupiter analogs. Following our previous work on this subject (Wittenmyer et al., 2011), we chose those stars which have, at present, at least 30 epochs over a time baseline of at least 3000 days. We then excluded those stars which did not satisfy the criterion $(RMS/\sqrt{N})\raisebox{-2.58334pt}{$\,\stackrel{{\scriptstyle\raisebox{-0.60275pt}{$\textstyle<$}}}{{\sim}}\,$}1$, an empirical relation derived from the AAPS data examined in Wittenmyer et al. (2011) which can be used as a simple estimate of whether the 12 m s^-1 velocity amplitude of Jupiter would be detectable at 99% confidence. After this cut, 103 stars remained, and all further analysis was done using those data.

The epochs of real observational data are never purely random: stars can only be observed at night, and most targets are unobservable for a portion of the year. When combined with the exigencies of telescope scheduling (planet-search programs are usually assigned time during bright lunations) real data can contain significant gaps. To better simulate the sampling characteristics of real data, we made the following assumptions: (1) one observation in a 10-night block every 30 days (bright-time scheduling), (2) the target is unobservable for four consecutive months every year, and (3) poor weather randomly prevents the observation 33% of the time. These conditions were selected for the following reasons: (1) planet-search programs are usually allocated time in bright lunations owing to the brightness of the targets, (2) for a mid-latitude site such as the Anglo-Australian Telescope (AAT), with planet-survey targets distributed randomly in right ascension, the average target is unobservable for four months in a year¹¹1Here we define “unobservable” to mean that the target spends less than one hour at an airmass less than 2., and (3) the long-term weather conditions at the AAT yield a 33% loss rate (this “weather allowance” is accounted for in the proposal process). Using this procedure, we added simulated observations to the current AAPS data (as of 2012 July) until reaching a desired number of added data points (chosen to lie between 9 and 24²²2These values were chosen to reflect the minimum (1.5 epochs/yr in 6 years) and maximum (8 epochs/yr for 3 years) practical cadences for a typical shared large telescope). In the interest of informing strategic plans for planet-search programs, we investigated the effect of adding observations over two timescales: three and six years. The frequency of observations (e.g. monthly or alternate months) was adjusted so that the resulting total time baseline of new data remained approximately constant at three and six years. Figure 1 shows the time baseline added using this approach, averaged over the 103 stars. In total, 32 artificial data sets were created for each target considered here: $N=9-24$ points added over 3 and 6 year periods. The radial velocity of the simulated data point is drawn from a Gaussian distribution with a FWHM equal to the RMS of the existing data for that star. The uncertainty on the simulated radial velocity is drawn at random (with replacement) from the uncertainties of the existing real data. The result is a simulated future data set with the same properties as the original data.

Throughout this work, we assessed the detectability of radial-velocity signals using the algorithm described extensively in our previous work (Wittenmyer et al., 2006, 2011). Here we only consider circular orbits (after e.g. Cumming et al. 2008), as previous work has shown that circular-orbit detection limits are a good approximation to the result that would be obtained for planets with $e\raisebox{-2.58334pt}{$\,\stackrel{{\scriptstyle\raisebox{-0.60275pt}{$\textstyle<$}}}{{\sim}}\,$}$0.5 (Cumming & Dragomir, 2010; Wittenmyer et al., 2010, 2011). In brief, for each of 100 trial periods, a Keplerian signal is added to the data, then a Lomb-Scargle periodogram (Lomb, 1976; Scargle, 1982) is used to search for the injected signal. The radial velocity semi-amplitude $K$ of the artificial planet is increased until 100% of the signals are recovered with at least 99.9% significance at the correct period. That value of $K$ is then the detection limit for that trial period. As we are concerned with long-period planets, we only consider periods from 1000-5000 days, evenly distributed in log space. Throughout this work, the figure of merit used to assess the impact of added data is the mean detection limit $K$ averaged over 100 trial periods in the range 1000-5000 days.

The most obvious question here is: “How does the number of added epochs affect the detectability of Jupiter analogs?” A related issue is whether there is a point of diminishing return, where adding more data or more frequent observations does not make meaningful improvements in the overall detectability. To determine the improvement to be had by various levels of observational effort, we also computed the detection limits for the 103-star sample exactly as above, but using only the existing AAPS data. This gives a “benchmark” detectability for each star, against which we can then compare the results from the simulated observations on a star-by-star basis. For each star, we divide the mean $K$ limit (averaged over periods from 1000-5000 days) by that obtained for the “benchmark” or present data set. The grand means of these ratios (averaged over the 103 stars) are plotted in Figure 2. From this figure, we can answer two questions: (1) What is the effect of adding epochs (within a fixed time), and (2) What is the impact of the observational baseline for a given number of epochs? Our results show that, for the range of added epochs simulated here, there remains a linear relationship between the number of epochs and the achievable detection limit. That is, the points shown in Figure 2 do not level off, up to 24 added epochs. On the second question, we see that there is no statistically significant difference between $N$ epochs added over 3 years and $N$ epochs added over 6 years, though this applies only for planets with $P<5000$ days (14 years) as this is the maximum period considered in the detection-limit calculation.

The results of this experiment, in which we added simulated observations to real AAPS data, show that the amount of added time coverage is less important than the number of added observations. It is worth noting here that the selected stars all had at least 3000 days (8 years) of time coverage before adding new data. Adding about 1000 or 2000 days (3 or 6 years) of new data thus has little effect on the detectability of Jupiter-like planets in the 1000-5000 day period range. We also note that the total number of epochs for each star varies greatly, with some stars at present having $\sim$35 epochs, and others having well over 100.

Since this could have biased the results shown in the Figure 2, we ran a second suite of simulations with slightly different initial conditions – less “realistic” but more controlled. We created wholly artificial data sets with a fixed number of observations $N$, ranging from 40 to 500 epochs. The time sampling was done exactly as described above, with the natural result that data sets with more observations had a longer timespan. The average time coverage of the simulated data sets ranged from 7.2 years ($N=40$) to 92 years ($N=500$). We have chosen the minimum $N$ as 40 epochs, which yields a time baseline of 7.2$\pm$0.7 years, consistent with the initial selection of AAPS stars used in the first suite of simulations. Also, in recognition of the fact that radial-velocity noise is almost certainly not Gaussian in nature (O’Toole et al., 2009a, b; Tuomi et al., 2013), we chose instead to draw the value of the simulated data point (and its uncertainty) from a “master” set of 531 observations from a selection of known stable stars in the AAPS catalog. Those velocities have an RMS of 3.0 m s^-1, which is typical of the long-term accuracy and precision achieved by AAT/UCLES over the lifetime of the AAPS. In this way, we assure that the simulated data are purely noise, with the characteristics of velocity noise from actual observed stars. The result is 100 simulated data sets each with $N$ epochs, for $N$ between 40 and 500. We computed the detection limits as described above, and calculated the mean velocity amplitude $K$ averaged over 100 trial periods for each data set, again averaging over periods from 1000-5000 days. Trial periods for which all simulated planets were not detected were ignored. Table 1 and Figure 3 show the results, where we have averaged the mean $K$ from each data set over all 100 sets at each value of $N$.

We see from Table 1 and Figure 3 that the detection limit $K$ scales approximately as $\sqrt{N}$, with the greatest marginal improvement occurring when the total duration extends beyond $\sim$4000 days (i.e. for $N=40-60$). This is the expected result for data consisting of Gaussian (white) noise (e.g. Cochran & Hatzes 1996); for the purposes of long-period planet detection, the radial-velocity noise distribution may be closer to Gaussian than is commonly expected. Up to the limit of our tests, $N=500$, no noise floor is reached - the detectabilities continue to improve as $\sqrt{N}$. We emphasize that this result applies for planets in the period range we have tested here: between 1000 and 5000 days. Obviously, continuing to observe a given star indefinitely would permit the detection of arbitrarily long periods shorter than the total duration of observations. The period dependence of our results is shown in Figure 4. Obviously, for the $N=50$ case, where the total time coverage is only about 9 years (3300 days), periods substantially longer than this baseline are virtually undetectable. This artifact disappears once the time coverage exceeds the 5000-day maximum trial period, and we see no significant period dependence for the larger-$N$ trials.

The mean detection limit drops below the 3.0 m s^-1 scatter of the input data after about 80 points (Figure 3, and reaches $K/\sigma\,=$0.5 when 300-350 epochs are obtained. Our result is applicable to other data sets with different noise properties – the detection limit would then simply scale with the overall noise. This can be seen from recent HARPS results, where the lowest-amplitude planetary signals are indeed at levels consistent with our results in Figure 3. For example, HD 10180b has $K/\sigma\,=$0.62 with $N=190$ measurements (Lovis et al., 2011). Similarly, Pepe et al. (2011) report HD 20794c with $K/\sigma\,=$0.68 and $N=187$ measurements. Because most radial-velocity planet search programs on shared large telescopes have similar sampling constraints, our results can thus be applied to other data sets by scaling the noise accordingly.