The evolution of hot Jupiters revealed by the age distribution of their host stars

Significance

Hot Jupiters are the first exoplanet population discovered around main-sequence stars. However, their origin and evolution remain puzzled. Using a sample with kinematic properties derived from large surveys (e.g., Gaia, LAMOST), we characterize the kinematic ages of stars hosting hot and warm/cold Jupiters, confirming the result of the previous study that hot Jupiter hosts are younger using a relative proxy. Furthermore, we find that as stars age, the frequency of hot Jupiters declines but that of warm/cold Jupiters does not vary. Such trends are expected from tidal decays of hot Jupiters’ orbits. Our derived relations can help explain the long-standing discrepancy between hot Jupiter frequencies from RV and transit surveys and the null detection of hot Jupiters in globular clusters.

Abstract

The unexpected discovery of hot Jupiters challenged the classical theory of planet formation inspired by our solar system. Until now, the origin and evolution of hot Jupiters are still uncertain. Determining their age distribution and temporal evolution can provide more clues into the mechanism of their formation and subsequent evolution. Using a sample of 383 giant planets around Sun-like stars collected from the kinematic catalogs of the Planets Across Space and Time project, we find that hot Jupiters are preferentially hosted by relatively younger stars in the Galactic thin disk. We subsequently find that the frequency of hot Jupiters declines with age as $F_{\mathrm{HJ}}\propto \exp (-0.20\pm 0.06\times \frac{t}{\mathrm{Gyr}})$. In contrast, the frequency of warm/cold Jupiters shows no significant dependence on age. Such a trend is expected from the tidal evolution of hot Jupiters’ orbits, and our result offers supporting evidence using a large sample. We also perform a joint analysis on the planet frequencies in the stellar age-metallicity plane. The result suggests that the frequencies of hot Jupiters and warm/cold Jupiters, after removing the age dependence are both correlated with stellar metallicities as $F_{\mathrm{HJ}}\propto {10}^{1.6_{-0.3}^{+0.3}\times [\mathrm{Fe}/\mathrm{H}]}$ and $F_{\mathrm{WJ}/\mathrm{CJ}}\propto {10}^{1.1_{-0.3}^{+0.2}\times [\mathrm{Fe}/\mathrm{H}]}$, respectively. Moreover, we show that the above correlations can explain the bulk of the discrepancy in hot Jupiter frequencies inferred from the transit and radial velocity (RV) surveys, given that RV targets tend to be more metal-rich and younger than transits.

Hot Jupiters generally refer to Jupiter-size planets with orbital periods $\lesssim$ 10 d around host stars. The existence of Jovian planets on such short periods poses significant challenges to the classical planet formation theories based on our solar system. Since the discovery of the prototype 51 Pegasi b in 1995 (1), hot Jupiters have been one of the most studied exoplanet populations, with a number of fascinating properties uncovered. For example, in contrast to the coplanar planetary orbits in the solar system, the orbits of some hot Jupiters are found to be inclined with respect to their stars’ equators [see a recent review by Albrecht et al. (2)]. Statistical works have explored how hot Jupiters distribute and their dependence on the properties of host stars. Remarkably, hot Jupiters are preferentially found around metal-rich stars (3, 4). Despite efforts over nearly three decades, the origin of hot Jupiters remains puzzling [see the review by Dawson and Johnson (5)].

An understudied probe into the formation and evolution history of hot Jupiters is their dependence on the hosts’ ages: Are there any differences in the hosts’ age distributions between hot Jupiters and their counterparts at longer periods (i.e., warm/cold Jupiters)? Does the frequency of hot Jupiters evolve with age? And if so, how? The answers to these questions can constrain their origin, especially revealing whether/how the tidal interactions with host stars shape their orbits.

The main bottleneck of investigating the temporal evolution of hot Jupiters has been the difficulty of making decent stellar age estimates for the hosts. The commonly used isochrone fitting method has a typical age uncertainty exceeding $\sim 50\%$ (e.g., ref. 6) for an individual main-sequence host. Alternatively, the velocity dispersion for an assemble of stars is known to correlate with age. The velocity dispersions have been previously used as a relative age proxy to study the hosts of hot Jupiters (7–9). Specifically, Hamer & Schlaufman found that the hosts of hot Jupiters are on average younger than the field stars, which can be interpreted by the tidal inspiral of hot Jupiters around hosts with modified stellar tidal quality factor $Q_{\ast }^{{}^{\prime }}\lesssim {10}^7$ (8). In a subsequent work, they showed that hot Jupiter host stars with larger obliquities are older compared to the aligned systems, suggesting those misaligned hot Jupiters arrived at their presently short-period orbits at late times (9).

The average age of an assemble of stars can be estimated statistically from their kinematics using the Age–Velocity dispersion Relation (AVR) (e.g., refs. 10 and 11). In the first two papers of the Planet Across Space and Time series (hereafter referred to as PAST I and II; refs. 12 and 13), we have refined the AVR to derive the kinematic ages with inner uncertainties of $\sim 10\%-20\%$ and constructed catalogs of stellar kinematic properties (e.g., Galactic position, velocity, and component membership) by combining data from the LAMOST, Gaia, APOGEE, RAVE, Kepler, and NASA Exoplanet Archive.^* In this work, benefiting from the kinematic methods and catalogs of the PAST series, we perform a statistical investigation into the age distribution and temporal evolution of hot Jupiters. Specifically, we aim to characterize the kinematic ages of stars hosting hot Jupiters and warm/cold Jupiters and deriving the frequency of hot Jupiters as functions of stellar age and metallicity.

Sample Selection

We collect our planet host stellar sample by selecting Sun-like stars in the Galactic disk from the planet host stellar catalog of PAST I (12). We then cross-match with the catalogs of confirmed planets and the Kepler DR 25 candidates from NASA exoplanet archive (https://exoplanetarchive.ipac.caltech.edu; ref. 14) and select giant planets as our planetary sample. Our sample contains 355 stars hosting 383 giant planets (SI Appendix, section 1—Table S1). Among these planets, 193 are hot Jupiters. In our sample, giant planets are discovered by various facilities (ground/space), with radial velocity (RV) or transit methods. In specific, our planetary sample consists of 29 hot Jupiters and 40 warm/cold Jupiters detected by space-based facilities with transit method (ST), 147 hot Jupiters, and 3 warm/cold Jupiters detected by ground-based facilities with transit method (GT), 17 hot Jupiters and 147 warm/cold Jupiters detected with RV method (RV) (SI Appendix, section 1—Table S2).

In SI Appendix, we provide more details on the sample selection and show how we characterize the age distribution of host stars and derive the evolution of the planetary frequency. Specifically, the procedure of sample selection is summarized in SI Appendix, section 1, and the mass-period distribution of selected giant planets is shown in SI Appendix, Fig. S1. The method to obtain the stellar age is described in SI Appendix, section 2. In SI Appendix, section 3, we provide detailed introductions on how to derive the planetary frequencies and their correlations with the stellar age/metallicity. We further discuss the implication of our results in SI Appendix, sections 4 and 5.

Analyses and Results

Hot Jupiters Are Preferentially Hosted by Younger Stars in the Thin Disk.

We first compare the kinematic properties of the hot Jupiter host stars to warm/cold Jupiter host stars. Fig. 1 displays their Toomre diagrams colored by the relative probabilities between thick disk to thin disk $TD/D$. As can be seen, compared to warm/cold Jupiter hosts, hot Jupiter hosts have smaller total velocities $V_{\mathrm{tot}}$ and $TD/D$. We perform the two-sample Kolmogorov–Smirnov (K-S) tests to evaluate the significance. As shown in the Bottom panels of Fig. 1, the resulting $P$-values are $0.035$ and $1.55\times {10}^{-7}$ for the $V_{\mathrm{tot}}$ and $TD/D$. We also resample the observed velocities from the normal distribution for 1,000 times and perform the two sample K-S tests for the resampled data. Out of the 1,000 times of resampled data, the resulted $P$-values of $V_{\mathrm{tot}}$ and $TD/D$ are less than 0.05 and 0.003 for 971 and 999 times, respectively, suggesting that hot Jupiter host stars statistically tend to belong to the thin disk with smaller Galactic velocity compared to the warm/cold Jupiter hosts.

Fig. 1.

Top panels: The Toomre diagrams color-coded by the relative probabilities of thick disk (TD) over thin disk (D), $TD/D$ for the hot Jupiter host stars (Top-Left panel) and warm/cold Jupiter host stars (Top-Right panel). Dotted lines show constant values of the total Galactic velocity $V_{\mathrm{tot}}={(U_{\mathrm{LSR}}^2+V_{\mathrm{LSR}}^2+W_{\mathrm{LSR}}^2)}^{1/2}$ in steps of 50 $\mathrm{k}m{s}^{-1}$. Bottom panels: The cumulative distributions of the total velocities $V_{\mathrm{tot}}$ (Bottom-Left panel) and the relative probabilities between thick disk (TD) to thin disk (D), $TD/D$ (Bottom-Right panel) for the hot Jupiter host stars (red) and warm/cold Jupiter host stars (blue). The two-sample K-S test $P$-values are plotted at the lower right corner of each panel. To evaluate the significance of the differences in $V_{\mathrm{tot}}$ and $TD/D$ between hot Jupiter hosts and warm/cold Jupiter hosts, we also plotted the cumulative distributions of the 1,000 sets of resampled data considering their uncertainties in light red/blue colors.

Previous work (8) using Gaia DR2 data found that the velocity dispersion of stars hosting hot Jupiter had smaller velocity dispersion than the field population, implying the former had a younger age on average. We calculate the average kinematic ages from the vertical velocity dispersions of hot Jupiter and warm Jupiter hosts with the refined AVR (SI Appendix, section 2.1). The resulting average ages are $3.{24}_{-0.32}^{+0.43}$ Gyr and $5.{21}_{-0.60}^{+0.68}$ Gyr for the hot Jupiter hosts and warm/cold Jupiter hosts, respectively (see the solid points in Fig. 2). Hot Jupiter host stars are $1.{97}_{-0.72}^{+0.75}$ Gyr younger than the warm/cold Jupiter host stars. In order to evaluate the significance of the age difference, we resample the data (i.e., Galactic velocities and coefficients of AVR) from their uncertainties and recalculate the kinematic ages. We find that the average kinematic ages of hot Jupiter host stars are younger 9,972 times out of 10,000 sets of resampled data, corresponding to a confidence level of 99.72%. It is worth noting that the kinematic age represents the average age of a group of stars. To further verify the above result, we also consider a subsample of MSTO stars having relatively well-determined individual ages with a typical uncertainty for an isochrone age of $\sim 20\%-30\%$ (e.g., 17, 18) (SI Appendix, section 2.2). As shown in Fig. 2, for the MSTO subsample, hot Jupiter hosts generally have younger isochrone ages compared to warm/cold Jupiter hosts. The average isochrone age of hot Jupiter hosts ($4.{05}_{-0.56}^{+0.81}$ Gyr) is smaller than that of warm/cold Jupiter hosts ($5.{60}_{-0.84}^{+0.78}$ Gyr) by $\sim 1.55$ Gyr, which is consistent with the kinematic results from the MSTO subsample, i.e., $3.{72}_{-0.42}^{+0.55}$ Gyr for hot Jupiters and $6.{22}_{-0.81}^{+0.97}$ Gyr for warm/cold Jupiters. In addition, we also compare the age distributions for hot Jupiter hosts and warm/cold Jupiter hosts from other sources (SI Appendix, Fig. S4), which show similar results.

Fig. 2.

The distributions of isochrone ages for the main-sequence turn-off (MSTO) subsamples of hot Jupiter hosts (dotted red histogram) and warm Jupiter hosts (dotted blue line histogram). For comparisons, the average kinematic ages of the hot Jupiter and warm/cold Jupiter hosts in the MSTO subsample and the whole sample are plotted as hollow points and solid points, respectively. As can be seen, the MSTO subsample is systematically older than the whole sample as expected, and hot Jupiter hosts are older than warm/cold Jupiter hosts in all the cases.

In SI Appendix, section 6, we discuss the influences of other stellar properties (e.g., $[Fe/H]$ and phase space density, SI Appendix, Figs. S29–S31) and show that they cannot (mainly) account for the above-mentioned observed age differences. Based on the above analyses, we conclude that hot Jupiters around Sun-like stars are preferentially hosted by younger stars in the Galactic thin disk.

The Frequency of Hot Jupiters as a Function of Age.

The above age difference implies that the frequency of hot Jupiters evolves with age. However, this result derived from planet hosts does not consider the detection efficiencies of field stars with null detection (which are the majority of survey targets) and may lead to biased conclusions. Therefore, in this section, we further correct the detection biases and explore the temporal evolution of the frequencies (i.e., the intrinsic fractions of stars hosting planets) of hot Jupiters and warm/cold Jupiters. As mentioned before, our sample consists of three subsamples, i.e., RV, GT, and ST. We construct the parent stellar samples for the three subsamples by using data from the Lick Planet Search, the LCES HIRES/Keck Precision Radial Velocity Exoplanet Survey, the public HAPRS RV database, Tycho-2/SuperWASP, and Kepler, respectively (SI Appendix, section 3.1.1, 3.2.1, and 3.3.1; refs. 19–26). For each subsample, we divide the giant planets and corresponding parent stars into several bins according to their kinematic ages and calculate the frequencies of hot Jupiters ($F_{\mathrm{HJ}}$) and warm/cold Jupiters ($F_{\mathrm{WJ}/\mathrm{CJ}}$) in each bin by correcting the geometric effect and detection efficiency (SI Appendix, sections 3.1.2, 3.2.2, and 3.3.2, and Figs. S5 and S11). SI Appendix, Figs. S6, S12, and S19 show $F_{\mathrm{HJ}}$ and $F_{\mathrm{WJ}/\mathrm{CJ}}$ as a function of average kinematic age derived from RV, GT, and ST subsamples, respectively.

The kinematic age is known to be correlated with other stellar properties such as mass and metallicities (e.g., refs. 27 and 22), which can also affect the frequency of giant planets (e.g., refs. 15, 16, and 28). Since we only study Sun-like hosts, the stellar masses in different age bins do not differ significantly, while the stellar metallicities $[Fe/H]$ decrease with increasing age (SI Appendix, Fig. S7). To qualify the dependence of $F_{\mathrm{HJ}}$ and $F_{\mathrm{WJ}/\mathrm{CJ}}$ on stellar age/$[Fe/H]$, we perform the Bayesian analysis for each of the three subsamples by modeling the planet frequency using three models: hybrid model $[Fe/H]$ ($F([\mathrm{Fe}/\mathrm{H}],t)=C\times {10}^{\beta [Fe/H]}\times \exp (\gamma \times t)$), single-age model ($F([\mathrm{Fe}/\mathrm{H}],t)=C\times \exp (\gamma \times t)$, i.e., $\beta =0$), and single-metallicity model ($F([\mathrm{Fe}/\mathrm{H}],t)=C\times {10}^{\beta [Fe/H]}$, i.e., $\gamma =0$) (SI Appendix, sections 3.1.3, 3.2.3, and 3.3.3). We also make a joint fit for the coefficients $\beta$ and $\gamma$ by combining the data from the three subsamples (note that we focus on deriving the relative frequency as functions of age and metallicity, and the normalization factors $C$ for all subsamples are set as free parameters; see SI Appendix, section 3.4). The hybrid model is preferred comparing to the single-age model and single-metallicity model with smaller Akaike information criterion (AIC; ref. 29) scores for all the cases, demonstrating that the trends of $F_{\mathrm{HJ}}$ and $F_{\mathrm{WJ}/\mathrm{CJ}}$ are the combined effects of varying age and $[Fe/H]$. The best-fit parameters and their $1-\sigma$ intervals for two-parameter samples are displayed in SI Appendix, Fig. S8, S9, S13, S20, and S21 and summarized in SI Appendix, section S3—Table S4. The best fits of the hybrid model can be mathematically expressed as:

$$F_{\text{HJ}}\propto {10}^{({1.6}_{-0.3}^{+0.3}\times [Fe/H])}\times \text{exp}(-{0.20}_{-0.06}^{+0.06}\times \frac{t}{\text{Gyr}}),$$ [1]

$$F_{\text{WJ}/\text{CJ}}\propto {10}^{({1.1}_{-0.3}^{+0.2}\times [Fe/H])}\times \text{exp}({0.02}_{-0.03}^{+0.02}\times \frac{t}{\text{Gyr}}).$$ [2]

Fig. 3 shows the marginalized posterior probability density distributions of $F_{\mathrm{HJ}}$ and $F_{\mathrm{WJ}/\mathrm{CJ}}$ from the joint fit and each of the three subsamples. For each subsample, the frequencies of hot Jupiters and warm/cold Jupiters are positively correlated ($\beta >0$) with stellar metallicity $[Fe/H]$ at the $\gtrsim 3\sigma$ levels. Moreover, the dependence of $F_{\mathrm{HJ}}$ on stellar $[Fe/H]$ ($\beta \sim 1.5-1.8$) is stronger than that of $F_{\mathrm{WJ}/\mathrm{CJ}}$ ($\beta \sim 1.0-1.2$). That is to say, giant planets, especially hot Jupiters, tend to be hosted by metal-richer stars, which is in general agreement with previous studies (15, 16).

Fig. 3.

Marginalized posterior probability density distributions for the metallicity and age exponents $(\beta ,\gamma )$ of the planet frequency function for hot Jupiters (Top panel) and warm/cold Jupiters (Bottom panel). The $1-\sigma$, $2-\sigma$, and $3-\sigma$ contours are displayed for RV, GT, and ST subsamples, and the joint fit in brown, purple, green, and black, respectively. The solid pentagrams denote the best-fit values of the joint fit. The cyan lines and regions represent the best-fit parameters and uncertainties of $\beta$ from previous studies (15, 16).

To isolate the effect of stellar age, in each age bin, we calculate the planet frequency normalized to the solar metallicity (i.e., $[Fe/H]=0$) according to the best-fits of $\beta$ (SI Appendix, section 3.1.3). Fig. 4 shows the metallicity-“corrected” frequencies of hot Jupiters $F_{\mathrm{HJ}}^{\mathrm{Cor}}$ and warm/cold Jupiters $F_{\mathrm{WJ}/\mathrm{CJ}}^{\mathrm{Cor}}$ for RV (brown), ST (green), and GT (purple), respectively. As can be seen, as kinematic age increases, the frequencies of hot Jupiters decline at confidence levels of 98.45%, 94.23%, 99.60%, and 99.99% for RV, GT, ST, and the joint fit, respectively. In comparison, warm/cold Jupiter frequencies are consistent with having no dependence on age as the fitted $\gamma$ for $F_{\mathrm{WJ}/\mathrm{CJ}}$ are consistent with zero within $1\sigma$.

Fig. 4.

The frequencies of hot Jupiters (Top panels) and warm/cold Jupiters (Bottom panels) as functions of average kinematic age for the ground-based RV (Left), space-based transit (Middle), and ground-based transit subsamples (Right), after normalizing the frequencies in each age bin to solar metallicity by applying the best-fit $\beta$. The solid lines and regions denote the best-fit models and $1-\sigma$ intervals. The mean values and the standard deviations in the respective bins are shown as solid circles with error bars. For the ground-based transit, we do not attempt to derive the absolute frequency (SI Appendix, section 3), and we renormalized the amplitude to the intersection of the best fits of ground-based RV and space-based transit for display purpose only.

Discussions and Conclusions

In this work, based on the giant planet sample selected from PAST I, we find that hot Jupiter host stars tend to be in the Galactic thin disk and are younger than warm/cold Jupiter hosts by $\sim 2$ Gyr on average (Figs. 1 and 2). Then, we derive the frequencies of hot Jupiters and warm/cold Jupiters as functions of stellar age and metallicities. We find that the frequency of warm/cold Jupiters shows no significant dependence on age while the frequency of hot Jupiters decreases with age significantly (Eqs. 1 and 2). Our sample shows the frequencies of hot Jupiters and warm/cold Jupiters exhibit positive correlations with stellar metallities, which are in qualitative agreement with previous studies. Nevertheless, the best-fit slope $\beta$ ($\sim$1.6) of $F_{\mathrm{HJ}}$ is somewhat shallower than that of previous work ($\sim$2.1; ref. 16), which is expected because the age dependence was not considered previously and thus $\beta$ was overestimated.

Age distribution of hot Jupiters could have important implications on their formation and evolution. There have been various proposed mechanisms (e.g., in situ formation, disk migration, and high-eccentricity migration) to form hot Jupiters and they operate on different timescales [see the review by Dawson and Johnson (5) and the references therein]. Specifically, in the cases of in situ formation and disk migration, hot Jupiters should be formed early before the gas disks dissipate (within $\lesssim 10$ Myr). In contrast, in the case of high-eccentricity migration, hot Jupiters could be delivered into $\lesssim 0.1$ AU through the whole lifetime of the host stars. The arrival timescales for hot Jupiters to reach within $\sim 0.1$ AU may impact how the observed hot Jupiter frequency depends on age: The observed hot Jupiter frequency declining with age may weaken the case for high-eccentricity migration to be the predominant channel if it could not deliver the majority of hot Jupiters in place at relatively early stage. In other words, our results imply that the bulk of hot Jupiters may arrive relatively early since the birth of their hosts because otherwise, the late-arrived hot Jupiter can lead to an increase in the hot Jupiter frequency hosted by relatively old stars (at least in certain age intervals).

Tidal effects (e.g., stellar equilibrium/dynamic tide, planetary equilibrium/obliquity tide) have been widely considered essential for the evolution of hot Jupiters’ short-period orbits (see, e.g., refs. 30 and 5, 31–34). The tidal inspiral timescale depends on the planetary mass, orbital period, stellar mass, radius, and tidal quality factor (see, e.g., refs. 30 and 32). Many previous studies have looked for observational evidence for the tidal decays from individual (e.g., the orbital decay of hot Jupiters; refs. 35–39) or ensemble properties of hot Jupiter systems (e.g., the observed distributions of orbital distance/eccentricity; refs. 40–43) as well as the properties (e.g., spin and Galactic velocity distributions) of hot Jupiter host stars (8, 44). Nevertheless, the tidal evolution of hot Jupiters remains uncertain. The stellar tidal quality factors derived from different works vary on several orders of magnitude (e.g., $\sim {10}^5$ from the tidal decay of WASP-12b and $\sim {10}^{7-8}$ from the distribution of assemble properties; refs. 36, 37, 42, 43). Due to tidal decay, some hot Jupiters may inject into the Roche limit within the stellar lifetimes and get tidally disrupted. Consequently, such a process could naturally result in a declining frequency of hot Jupiters as a function of age, like the observed trend found in our work. For warm/cold Jupiters at longer orbital periods, the tidal inspiral timescales are expected to be too long to play any (significant) role in their evolution, which agrees with our result that there is no observed change of their frequency with age. Moreover, one can even derive the stellar tidal factor by fitting the hot Jupiter frequency–age relation (Fig. 4 and Eq. 1) with tidal evolution models. Such analyses also rely on hot Jupiter formation mechanisms because different mechanisms operate on different timescales and they set the initial conditions for the tidal evolution afterward. In summary, our observational results can potentially be used to simultaneously constrain stellar tidal quality factors and test various formation models (to be explored in a follow-up work).

In the following, we discuss the implication of our results on addressing the discrepant frequencies of hot Jupiters derived from RV and transit surveys (see, e.g., ref. 51). The frequencies of hot Jupiters inferred from RV surveys ($\sim 0.9\%$–$1.5\%$; e.g., refs. 48–50) are significantly higher than those derived from transit surveys ($\sim 0.3\%$–$0.8\%$; refs. 45–47). Recently, some works have proposed that such a discrepancy could be potentially explained by the difference in the close binary fractions between RV and transit targets (52, 53). The close binaries may also influence warm Jupiters (giant planets within $\sim 0.1-1$ AU) if this is the case. Interestingly, there seems to be no significant difference in the frequencies of warm Jupiters between RV ($\sim 4\%$–$5\%$; refs. 54 and 55) and transit (e.g., $4.1\pm 1.2\%$ from LAMOST-Gaia-Kepler catalog; SI Appendix, section 4) surveys. Here, we investigate whether a combination of the metallicity and age effects could help address the frequency discrepancy. We compare the stellar and planetary properties of our RV and transit subsamples and find that 1) the metallicity of RV stars ($[Fe/H]\sim 0.02$) are on average metal-richer than transit stars ($[Fe/H]\sim -0.04$) (SI Appendix, Fig. S24); 2) hot Jupiters discovered by transit method are on average $\sim 1$ Gyr older than RV subsamples (SI Appendix, Figs. S25 and S26). By using our derived $F_{\mathrm{HJ}}$ function (Eq. 1) and taking the age and metallicity differences from the RV surveys into account, we derive a modified $F_{\mathrm{HJ}}$ ($\sim 1.1\%$) from Kepler survey, which is consistent with the frequencies derived from RV surveys within $\sim 1\sigma$ (as shown in Fig. 5; see detailed discussions in SI Appendix, section 4).

Fig. 5.

The frequencies of hot Jupiters $F_{\mathrm{HJ}}$. Case 1: Result derived in previous studies from transit surveys (e.g., refs. 45–47); Case 2: Observed $F_{\mathrm{HJ}}$ derived in this work based on the Kepler sample; Case 3: Modified $F_{\mathrm{HJ}}$ of the Kepler result by taking account of the $[Fe/H]$ difference from the RV sample; Case 4: Modified $F_{\mathrm{HJ}}$ of Kepler results by taking account of the $[Fe/H]$ and kinematic age differences from the RV sample; Case 5: Observed $F_{\mathrm{HJ}}$ derived in this work based on the RV sample; Case 6: Results derived in previous studies from RV surveys (e.g., refs. 48–50).

With the advantage of accurate stellar age and metallicity measurements, stellar clusters are attractive environments to search and study hot Jupiters. So far, searches in globular clusters composed of old stellar populations have yielded null detection, suggesting a lower frequency of hot Jupiters in globular clusters compared to those in field stars (56–59). In comparison, tens of hot Jupiters (candidates) have been discovered in open clusters containing young stars, and statistical studies mostly yield $F_{\mathrm{HJ}}$ higher than field stars after accounting for the effect of $[Fe/H]$ (SI Appendix, section 5 and Table S5; e.g., refs. 60–64). The above trends of hot Jupiter frequencies in globular and open clusters are in broad agreement with expected lower/higher frequencies of hot Jupiters with old/young stars from our results. In the following, we consider the null detection by Gilliland et al. from searching for hot Jupiters in the globular cluster 47 Tucanae. It was expected to detect $\sim 17$ hot Jupiters if assuming its hot Jupiter frequency to be the same as those in field stars, but zero was found (65). Masuda and Winn revisited the problem by using the hot Jupiter frequency measured in the Kepler field and limiting their stellar masses to be in the same range as those in 47 Tucanae, and they found that the expected detection number is significantly reduced to $2.2_{-1.1}^{+1.6}$ (59). We find that by further considering the effect of age, the frequency of hot Jupiters in 47 Tucanae (age of $\sim$11.6 Gyr) will decrease by another factor of $\sim$2.7, and the expected number of detected hot Jupiters ($0.8_{-0.5}^{+0.7}$) is thus in little tension with the observed null result (SI Appendix, Fig. S28).

Future discoveries and analysis of hot Jupiters (e.g., refs. 66 and 67) may probe a broader range of stellar ages and metallicities in various stellar environments (e.g., star formation regions, stellar clusters, associations, halo; refs. 68 and 5), allowing for testing our results and offering better understanding on the age distribution and evolution of hot Jupiters.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.