Planets Across Space and Time (PAST). III. Morphology of the Planetary Radius Valley as a Function of Stellar Age and Metallicity in the Galactic Context Revealed by the LAMOST-Gaia-Kepler Sample

We initialized our stellar sample from the LAMOST-Gaia-Kepler catalog from PAST II (Chen et al. 2021b), a total of 35,835 Kepler stars, including stellar hosts of 1060 Kepler planets (candidates). In addition to some basic stellar parameters (e.g., mass, radius, effective temperature, [Fe/H], [α/Fe]), the catalog provides stellar kinematic properties, such as the Galactic velocities and the derived relative membership probabilities among different Galactic components (TD/D, TD/H, Herc/D, Herc/TD), where D, TD, Herc, and H denote the thin disk, the thick disk, the Hercules stream, and the halo of the Milky Way, respectively.

For the star sample considered in this paper, we first excluded giant stars by eliminating stars with ${\mathrm{log}}_{10}\tfrac{{R}_{* }}{{R}_{\odot }}\,\gt 0.00035\times ({T}_{\mathrm{eff}}-4500)+0.15$ according to Fulton et al. (2017). The stellar mass, radius, and T_eff are adopted from the Gaia-Kepler stellar properties catalog of Berger et al. (2020b). Following Bensby et al. (2014), we then adopted the criteria "Herc/D < 0.5 & Hec/TD < 0.5 & TD/H > 1" to keep only stars in the Galactic disk because kinematic age only applies to stars belonging to the Galactic disk components as suggested in PAST I (Chen et al. 2021a). Finally, we removed stars without [Fe/H] or [α/Fe] measurements.

For the planet sample, we excluded planets of grazing transits (i.e., $\tfrac{{R}_{p}}{{R}_{* }}+b\gt 1$, where b is the transit impact parameter in the Kepler DR25 table from the NASA exoplanet archive (NASA Exoplanet Archive 2021) and R _p is calculated by multiplying stellar radius and the radius ratio of planet to star) because their radius measurements generally suffer from large uncertainties. To ensure the detection efficiency of super-Earths, we only kept planets with orbital period less than 100 days. We also excluded ultra–short-period planets (USPs, planets with orbital period less than 1 day) because they are relatively rare and likely to be formed differently as compared to the bulk of Kepler planets (Winn et al. 2018). Further, since we mainly focus on planets close to the radius valley, we only kept planets with radii in the range of 1–6 R_⊕. After the above selections, we are left with 446 stars hosting 621 planets (candidates). Table 1 is a rundown of the above sample selection process.

Table 1. Rundown of the Sample Selection

Selection Criteria	N s	N p
LAMOST-Gaia-Kepler sample (PAST II)	764	1060
Excluding giant stars	747	1040
Belong to Galactic disks	659	919
With [Fe/H] and [α/Fe] measurements	605	850
$\tfrac{{R}_{p}}{{R}_{* }}+b\leqslant 1$	589	833
Period ≤ 100 days	544	767
Period ≥ 1 days	532	753
1 R⊕ < R p < 6 R⊕	446	621

Note. N _s and N _p are the numbers of host stars and planets, respectively, during the process of sample selection in Section 2.1.

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Figure 1 displays the color-coded distributions of relative probabilities between thick disk and thin disk (TD/D) in the [Fe/H] − [α/Fe] plane for our stellar sample. As can be seen, in general, stars with larger TD/D (kinematically thicker) have lower [Fe/H] and higher [α/Fe], which is consistent with previous studies (e.g., Bensby et al. 2014; Chen et al. 2021a).

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According to previous studies (e.g., Owen & Wu 2013, 2017; Fulton et al. 2017; Fulton & Petigura 2018; Zhu & Dong 2021), the radius valley is located at ∼1.9 ± 0.2 R_⊕. We then divide our planet sample into four subsamples according to their radii:

• 1.  
  Valley planet (VP): 1.7–2.1 R_⊕ (N_VP = 70).
• 2.  
  Super-Earth (SE): 1.0–1.7 R_⊕ (N_SE = 266).
• 3.  
  Sub-Neptune (SN): 2.1–3.5 R_⊕ (N_SN = 238).
• 4.  
  Neptune-size planet (NP): 3.5–6 R_⊕ (N_NP = 47).

Here N_VP, N_SE, N_SN, and N_NP are the numbers of planets in the corresponding subsamples.

Figure 2 shows the period–radius distributions of our planetary sample. As can be seen, there exists an obvious bimodal distribution in planetary radius and a valley in ∼1.9 R_⊕, which is well consistent with previous studies (e.g., Fulton et al. 2017). In the subsequent sections, we will further characterize the morphology of the radius valley and explore its dependence on various stellar properties.

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To characterize the morphology of the radius valley, we adopted a set of metrics, which are defined as follows:

• 1.  
  The contrast of radius valley C_valley, defined as the number ratio of super-Earths plus sub-Neptunes to the valley planets, i.e.,

  $$\begin{eqnarray}&&{C}_{\mathrm{valley}}=\displaystyle \frac{{N}_{\mathrm{SE}}+{N}_{\mathrm{SN}}}{{N}_{\mathrm{VP}}}.\end{eqnarray} \tag{ 1 }$$
• 2.  
  The asymmetry on the two sides of the valley defined as the number ratio (in logarithm) of super-Earths to sub-Neptunes, i.e.,

  $$\begin{eqnarray}&&{A}_{\mathrm{valley}}={\mathrm{log}}_{10}\left(\displaystyle \frac{{N}_{\mathrm{SE}}}{{N}_{\mathrm{SN}}}\right).\end{eqnarray} \tag{ 2 }$$
• 3.  
  The average radius of planets with size larger than valley planets, i.e.,

  $$\begin{eqnarray}&&{R}_{\mathrm{valley}}^{+}=\frac{1}{{N}_{\mathrm{SN}}+{N}_{\mathrm{NP}}}\displaystyle \sum _{i}^{{N}_{\mathrm{SN}}+{N}_{\mathrm{NP}}}{R}_{i},\end{eqnarray} \tag{ 3 }$$

  where 2.1 R_⊕ <R _i < 6 R_⊕.
• 4.  
  The average radius of planets with size smaller than valley planets, i.e.,

  $$\begin{eqnarray}&&{R}_{\mathrm{valley}}^{-}=\frac{1}{{N}_{\mathrm{SE}}}\displaystyle \sum _{i}^{{N}_{\mathrm{SE}}}{R}_{i},\end{eqnarray} \tag{ 4 }$$

  where 1.0 R_⊕ <R _i < 1.7 R_⊕.
• 5.  
  The number fraction of Neptune-size planets in the whole planet sample, i.e.,

  $$\begin{eqnarray}&&{f}_{\mathrm{NP}}=\displaystyle \frac{{N}_{\mathrm{NP}}}{{N}_{\mathrm{SE}}+{N}_{\mathrm{VP}}+{N}_{\mathrm{SN}}+{N}_{\mathrm{NP}}}.\end{eqnarray} \tag{ 5 }$$

  This metric reflects the extension degree of the second (with larger radius) peak beside the radius valley.

To obtain the uncertainties of these metrics, following Berger et al. (2020a), we performed Monte Carlo simulations by resampling the planet radii from a normal distribution given their measured values and uncertainties. We then recounted N_VP, N_SE, N_SN, and N_NP and computed those metrics, i.e., A_valley, C_valley, R_valley ⁺, ${R}_{\mathrm{valley}}^{-}$, and f_NP, for the resampled data. We repeated the above procedure 10,000 times and then calculated the 84.1 percentiles and 15.9 percentiles in the resample distributions as the upper and lower uncertainties of the corresponding metrics.

In this paper, we only considered the Galactic disk components, i.e., thin and thick disks, and used the relative probability TD/D to quantify the likelihood that a star belongs to the thick disk relative to the thin disk. The higher the TD/D is, the more likely the star is to belong to the thick disk than the thin disk. We investigated the radius valley morphology as a function of TD/D. To do this, we divided our sample into four bins according to TD/D. Specifically, we first sorted the whole sample in the order in which TD/D increases. We took 10% (44) of stars at the lower TD/D end as the first bin and divided the rest into three bins of equal size (134) according to their TD/D. To remove the effects of other stellar parameters and planet detection efficiency on different bins, the last three bins were further controlled to let them have similar distributions in stellar mass, radius, and photometric precision (i.e., combined differential photometric precision (CDPP)) as compared to the first bin. This was realized by adopting the NearestNeighbors method in scikit-learn (Pedregosa et al. 2011) to select the nearest neighbor in the space of the controlled parameters from stars in the last three bins for every star in the first bin (see Appendix A for the details of the construction and validation of the control bins). After the above parameter control process, the last three bins have similar distributions in stellar mass, radius, and CDPP as compared to the first bin (Kolmogorov–Smirnov (K-S) test P-value greater than 0.8 as shown in Figure A2). We stress that this method removes the difference in sample completeness between bins, but it does not derive the intrinsic distribution in each bin by directly applying completeness corrections, so our statistical results are meaningful in the differential sense among various bins. Note that here we did not control [Fe/H] and [α/Fe] because the differences in stellar metallicity between thin- and thick-disk stars are essential and inevitable.

Figure 3 displays the period–radius diagram (top panel) and the radius distribution (bottom panel) of planets in different TD/D bins (after parameter control). As expected and printed at the top of the figure, with the increase of TD/D (i.e., from thin disk to thick disk), the kinematic age and [α/Fe] increase, while the [Fe/H] decreases. However, there seems no apparent pattern between radius valley and TD/D. Such an intuition is confirmed in Figure 4, in which we plot the five metrics (C_valley, A_valley, ${R}_{\mathrm{valley}}^{+}$, ${R}_{\mathrm{valley}}^{-}$, and f_NP, Equations (1)–(5)) of radius valley morphology as a function of TD/D. As can be seen, with the increase of TD/D, C_valley and A_valley seem to first increase and then decrease, while ${R}_{\mathrm{valley}}^{-}$ and f_NP first decrease and then increase. We have also adjusted the size of the first bin (e.g., 15%) and found that the evolution trends of the five metrics of radius valley morphology generally maintain, demonstrating that our results are not (significantly) dependent on the selection of first bins.

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However, there is no clear monotonic trend between TD/D and any of the morphology metrics owing to such small number statistics. Such a nonmonotonic result is not unexpected because thick-disk stars are intrinsically older with lower (higher) [Fe/H] ([α/Fe]) compared to thin-disk stars. Thus, the results of increasing TD/D are essentially a combination of the effects of growing age, decreasing [Fe/H], and increasing [α/Fe]. In the following subsections, we will investigate these effects separately.

In this subsection, we focus on the effect of age on the radius valley morphology. Similar to the parameter control method in Section 3.1, we further control [Fe/H] and [α/Fe] to isolate the effect of age as shown in Figure A3.

We estimated the average age of each bin with the kinematic method by using the age–velocity dispersion relation (AVR) refined by PAST I (Chen et al. 2021a). In Figure 5, we show the period–radius diagram (top panel) and radius distribution (bottom panel) of planets in each bin. The median values and 1σ uncertainties of kinematic age, [Fe/H], and [α/Fe] of each bin are printed at the top. From left to right, the age increases monotonically and significantly, while [Fe/H] and [α/Fe] are almost unchanged among different bins, which is expected after the above parameter control. Some apparent age trends emerge in Figure 5. As can be seen, the radius valley looks more prominent in older bins, and the number ratio of super-Earths to sub-Neptunes seems to increase with age. In addition, the radius range of planets above the radius valley is shrinking over the age.

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We further investigate the above apparent trends quantitatively by calculating the five radius morphology metrics (Equations (1)–(5)) in each bin. Figure 6 shows these metrics, i.e., C_valley, A_valley, ${R}_{\mathrm{valley}}^{+}$, ${R}_{\mathrm{valley}}^{-}$, and f_NP, as a function of age from the top to bottom panels. For each panel in Figure 6, we fit the data points with two models, i.e., a constant model (y = C) and a linear model with a logarithm timescale ($y=A\times {\mathrm{log}}_{10}(t)+B$), or in log–log space (${\mathrm{log}}_{10}(y)=A\times {\mathrm{log}}_{10}(t)+B$). We adopted two approaches to evaluate the two models. First, the simplest one is to calculate the Akaike information criterion (AIC) score of the constant and linear models, AIC_con and AIC_lin. In terms of the residual sum of squares (RSS), the AIC can be calculated with the following formula (Cavanaugh 1997):

$$\begin{eqnarray}&&\mathrm{AIC}=2k+n\mathrm{ln}(\mathrm{RSS}),\end{eqnarray} \tag{ 6 }$$

where n is the number of data points and k is the number of model parameters. We then calculate the difference in the AIC scores (ΔAIC ≡ AIC_con − AIC_lin) of the two models, and the model with the smaller AIC is preferred. Second, we adopted a Monte Carlo simulation by resampling the data points given their uncertainty and refitting the data 10,000 times. The confidence level of the best-fit parameter is calculated as the fraction of these resample-fit trials in which the model has a lower AIC score. The fitting results are as follows:

• 1.  
  For C_valley, the linear increasing model is preferred over the constant model with a ΔAIC = 7.65 and a confidence level of 95.21% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{C}_{\mathrm{valley}}={8.6}_{-3.4}^{+2.5}\times {\mathrm{log}}_{10}t+{7.7}_{-1.5}^{+0.4}.\end{eqnarray} \tag{ 7 }$$
• 2.  
  For A_valley, the linear increasing model is preferred over the constant model with a ΔAIC = 5.66 and a confidence level of 96.57% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{A}_{\mathrm{valley}}={0.20}_{-0.03}^{+0.08}\times {\mathrm{log}}_{10}t+{0.10}_{-0.03}^{+0.01}.\end{eqnarray} \tag{ 8 }$$

  Although we simply fit a linear function here owing to limited data points, we note that the increase of A_valley (or the number ratio of super-Earths to sub-Neptunes equivalently) seems nonlinear; it is mainly completed within ∼1–2 Gyr (see more discussions in Section 4.2.1).
• 3.  
  For ${R}_{\mathrm{valley}}^{+}$, the linear decay model is preferred over the constant model with a ΔAIC = 12.60 and a confidence level of 99.71% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{\mathrm{log}}_{10}\left(\displaystyle \frac{{R}_{\mathrm{valley}}^{+}}{{R}_{\oplus }}\right)=-{0.08}_{-0.01}^{+0.01}\times {\mathrm{log}}_{10}t+{0.46}_{-0.05}^{+0.04}.\end{eqnarray} \tag{ 9 }$$
• 4.  
  For ${R}_{\mathrm{valley}}^{-}$, the constant model is preferred over the linear model with a ΔAIC = 2.13 and a confidence level of 99.32% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{R}_{\mathrm{valley}}^{-}={1.34}_{-0.16}^{+0.04}{R}_{\oplus }.\end{eqnarray} \tag{ 10 }$$
• 5.  
  For f_NP, the linear decreasing model is preferred over the constant model with a ΔAIC = 10.12 and a confidence level of 99.85% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{f}_{\mathrm{NP}}=-{0.10}_{-0.02}^{+0.01}\times {\mathrm{log}}_{10}t+{0.08}_{-0.02}^{+0.02}.\end{eqnarray} \tag{ 11 }$$

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Above quantitative results confirm the apparent trends seen in Figure 5. With increasing age over gigayear timescales, the radius valley becomes more prominent (i.e., larger C_valley) and more asymmetric (larger A_valley or larger number ratio of super-Earths to sub-Neptunes); the average radii of planets larger than valley planets decrease (with a slope of $d(\mathrm{log}{R}_{\mathrm{valley}}^{+})/d(\mathrm{log}t)\,\sim -0.07$), while the average radii of planets smaller than valley planets (i.e., super-Earth) changes little; and the fraction of Neptune-size planets f_NP decreases significantly. The implications of these results on planet formation and evolution will be discussed in Section 4.3.

In this subsection, we explore the dependence of radius valley on [Fe/H]. Similar to the parameter control method in Section 3.1, we controlled the parameters, i.e., stellar mass, radius, CDPP, [α/Fe], and TD/D, to isolate the effect of [Fe/H]. After the parameter control process, stars in different bins have similar distributions in these controlled parameters (with all K-S test p-values larger than 0.2) as shown in Figure A4. Note that we were not allowed to control stellar age directly. In fact, age has been controlled (see the ages printed at the top of Figure 7) indirectly by controlling TD/D because TD/D and age are strongly correlated to each other (Chen et al. 2021a).

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In Figure 7, we compare the period–radius diagram (top panel) and radius distribution (bottom panel) of planets in different [Fe/H] bins after parameter control. As printed at the top, the median values of [Fe/H] decrease continuously, while the kinematic age and [α/Fe] are nearly unchanged within their uncertainty, demonstrating that these parameters have been well controlled. Some apparent trends emerge in Figure 7. As can be seen, the radius valley seems to be filled up as [Fe/H] decreases. There seem to be more sub-Neptunes relative to super-Earths in the Fe-rich bins than in the Fe-poor bins. In addition, the radius range of planets above the radius valley is shrinking from Fe-rich bins to Fe-poor bins.

To quantify the above trends, we further calculated the five radius valley morphologies (Equations (1)–(5)) for the four bins of different metallicity [Fe/H]. In Figure 8, we show these metrics, i.e., C_valley, A_valley, ${R}_{\mathrm{valley}}^{+}$, ${R}_{\mathrm{valley}}^{-}$, and f_NP, as a function of [Fe/H] from the top to bottom panels. We fit the data points in each panel with two models, i.e., a constant model (y = C) and a linear model (y = A × [Fe/H] + B), or in a logarithm ordinate scale ${\mathrm{log}}_{10}(y)=A\times [\mathrm{Fe}/{\rm{H}}]+B$). Then, following the same procedures as in Section 3.2, we evaluated these two models by calculating the difference in AIC score (ΔAIC) and the confidence level from Monte Carlo simulation. The fitting results are as follows:

• 1.  
  For C_valley, the linear increasing model is preferred over the constant model with a ΔAIC = 10.13 and a confidence level of 95.72% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{C}_{\mathrm{valley}}={26.3}_{-8.8}^{+3.4}\times [\mathrm{Fe}/{\rm{H}}]+{10.4}_{-1.1}^{+1.5}.\end{eqnarray} \tag{ 12 }$$
• 2.  
  For A_valley, the linear decay model is preferred over the constant model with a ΔAIC = 10.51 and a confidence level of 99.92% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{A}_{\mathrm{valley}}=-{0.6}_{-0.1}^{+0.1}\times [\mathrm{Fe}/{\rm{H}}]+{0.10}_{-0.02}^{+0.02}.\end{eqnarray} \tag{ 13 }$$
• 3.  
  For ${R}_{\mathrm{valley}}^{+}$, the linear increasing model is preferred over the constant model with a ΔAIC = 10.70 and a confidence level of 99.46% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{\mathrm{log}}_{10}\left(\displaystyle \frac{{R}_{\mathrm{valley}}^{+}}{{R}_{\oplus }}\right)={0.09}_{-0.03}^{+0.02}\times [\mathrm{Fe}/{\rm{H}}]+{0.46}_{-0.02}^{+0.06}.\end{eqnarray} \tag{ 14 }$$
• 4.  
  For ${R}_{\mathrm{valley}}^{-}$, the constant model is preferred over the linear model with a ΔAIC = 1.87 and a confidence level of 97.28% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{R}_{\mathrm{valley}}^{-}={1.33}_{-0.04}^{+0.05}{R}_{\oplus }.\end{eqnarray} \tag{ 15 }$$
• 5.  
  For f_NP, the linear increasing model is preferred over the constant model with a ΔAIC = 4.71 and a confidence level of 95.23% from Monte Carlo simulation. The best fit is

  $$\begin{eqnarray}&&{f}_{\mathrm{NP}}={0.18}_{-0.05}^{+0.04}\times [\mathrm{Fe}/{\rm{H}}]+{0.07}_{-0.01}^{+0.02}.\end{eqnarray} \tag{ 16 }$$

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Above quantitative results confirm the apparent trends seen in Figure 7. With the increase of [Fe/H] the radius valley becomes emptier; the ratio of super-Earths to sub-Neptunes decreases significantly; the average radii of planets larger than valley planets increases continuously with a slope of $d\mathrm{log}{R}_{\mathrm{valley}}^{+}/d[\mathrm{Fe}/{\rm{H}}]\sim 0.1$, while the average radius of smaller planets, i.e., super-Earth, is almost unchanged; and the fraction of Neptune-sized planets increases significantly. The implications of these results on the formation mechanisms of the radius valley will be discussed in Section 4.3.

In this subsection, we study the link between the radius valley and [α/Fe]. Similar to the parameter control method in Section 3.1, we controlled the parameters, i.e., stellar mass, radius, CDPP, [Fe/H], and TD/D, to isolate the effect of [α/Fe]. After the parameter control process, stars in different bins have similar distributions in these controlled parameters (with all K-S test p-values larger than 0.15) as shown in Figure A5.

Figure 9 shows the period–radius diagram (top panel) and radius distribution (bottom panel) of planets in different [α/Fe] bins after parameter control. As printed at the top, from left to right panels, the median values of [α/Fe] increase continuously, while the kinematic age and [Fe/H] are nearly unchanged given their uncertainties. We see an apparent decay in super-Earths accompanied by a rise in sub-Neptunes as [α/Fe] increases. This apparent trend is confirmed in the second panel of Figure 8, which shows the asymmetry of the valley A_valley as a function of [α/Fe]; a linear decay model is preferred over the constant model with a ΔAIC = 13.75 and a confidence level of 99.99% from Monte Carlo simulation. The best fit is

$$\begin{eqnarray}&&{A}_{\mathrm{valley}}=-{1.9}_{-0.1}^{+0.4}\times [\alpha /\mathrm{Fe}]+{0.03}_{-0.02}^{+0.02}.\end{eqnarray} \tag{ 17 }$$

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Except for A_valley, the other morphology metrics do not show a monotonic trend with [α/Fe]. The C_valley seems to be a constant function of [α/Fe]. For the other three metrics (i.e., ${R}_{\mathrm{valley}}^{+}$, ${R}_{\mathrm{valley}}^{-}$, and f_NP ), they all seem to follow a falling−rising V-shape pattern. However, we caution that this pattern could be unphysical because there are only four data points (four bins) here, while a V-shape model needs at least four parameters. Besides the small number of bins, the total range of [α/Fe] (between −0.1 and 0.1) is also small, and the differences in [α/Fe] between adjacent bins are only ∼1σ (see the horizontal error bars in Figure 9). All these limitations add difficulty to further revealing the physical connection between radius valley morphology and [α/Fe].

In all the above investigations, we eliminate the influence of the observational bias indirectly via constructing control samples. In Appendix B, we calculated the transit detection efficiency and found that the mean detection efficiency, or completeness, does not change substantially for our controlling samples across all TD/D, age, [Fe/H], and [alpha/Fe] (as shown in Figures 10–A3). Therefore, we conclude that completeness corrections are not likely to (significantly) impact the main results of our analysis. Since previous studies from observational analyses (e.g., Berger et al. 2020a; Sandoval et al. 2021) and theoretical models (e.g., Gupta & Schlichting 2020; Rogers & Owen 2021) all adopt the raw number ratio of super-Earths to sub-Neptunes, here we also adopt the raw number ratio to facilitate subsequent comparisons and discussions in Section 4.

In this paper, the radius valley is nominally and simply defined at ${R}_{{\rm{p}}0}^{\mathrm{valley}}=1.9\pm 0.2\,{R}_{\oplus }$. In fact, the center of the radius valley is found to be dependent on stellar mass (Wu 2019; Berger et al. 2020a) and planetary orbital period (Van Eylen et al. 2018), which can be characterized as

$$\begin{eqnarray}&&{R}_{p}^{\mathrm{valley}}={R}_{p0}^{\mathrm{valley}}{\left(\frac{{M}_{* }}{{M}_{\odot }}\right)}^{h}{\left(\frac{P}{10\ \mathrm{days}}\right)}^{g},\end{eqnarray} \tag{ 18 }$$

where P and M_* are planetary orbital period and stellar mass, respectively, and $h={0.26}_{-0.16}^{+0.21}$ (Berger et al. 2020a) and $g=-{0.09}_{-0.04}^{+0.02}$ (Van Eylen et al. 2018) are the corresponding slopes.

To take the above dependence into account, following Zhu & Dong (2021), we rescaled all planet radii as

$$\begin{eqnarray}&&\tilde{{R}_{p}}\equiv {R}_{p}{\left(\frac{{M}_{* }}{{M}_{\odot }}\right)}^{-h}{\left(\frac{P}{10\ \mathrm{days}}\right)}^{-g},\end{eqnarray} \tag{ 19 }$$

where h and g were set as 0.26 and −0.09, respectively. With the rescaled radii, then the boundaries (Section 2.1) used to classify different planet populations should be updated accordingly, i.e.,

• 1.  
  Valley planet (VP): $1.7\,{R}_{\oplus }\leqslant \tilde{{R}_{p}}\leqslant 2.1\,{R}_{\oplus };$.
• 2.  
  Super-Earth (SE): ${R}_{p}\geqslant 1\,{R}_{\oplus }\,\mathrm{and}\,\tilde{{R}_{p}}\lt 1.7\,{R}_{\oplus };$.
• 3.  
  Sub-Neptune (SN): $\tilde{{R}_{p}}\gt 2.1\,{R}_{\oplus }\,\mathrm{and}\,{R}_{p}\leqslant 3.5\,{R}_{\oplus };$.
• 4.  
  Neptune-size planet (NP): 3.5 R_⊕ < R _p ≤ 6 R_⊕.

Note that here the boundaries of the Neptune-size planets are not affected by the change in the valley boundaries.

With the above new criteria, we recalculated the five metrics of radius valley morphology, which are plotted as hollow gray circles in Figures 4, 6, 8 and 10, respectively. As can be seen, the recalculated C_valley, A_valley, f_NP, ${R}_{\mathrm{valley}}^{+}$, and ${R}_{\mathrm{valley}}^{-}$ (open circles) agree well with our nominal results (filled points). To quantitatively characterize the influence of stellar mass and planetary period on radius valley, we also fit those gray data points with the same procedure described in Section 3. The results are all consistent with our nominal results within 1σ uncertainty.

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Based on the above analysis, we therefore conclude that our results (e.g., Figures 4, 6, 8, and 10) are not (significantly) affected by the dependence of radius valley on stellar mass and planetary orbital period. Here we did not implement the rescaled radii from the beginning because the boundaries between sub-Earth and super-Earth, sub-Neptune, and Neptune should not be scaled. Once taking the rescaled radii, the above boundaries should be different for different planets and cannot be easily plotted in Figures 3, 5, 7, and 9.

4.2.1. Studies of Age Dependence

4.2.2. Studies of Metallicity Dependence

4.3.1. Radius Valley Emerged before Gyr and Evolving beyond Gyr

4.3.2. Constraints on Planetary Thermodynamic Evolution

4.3.3. Effects of Metallicity on Planetary Formation and Evolution