Exoplanets Around Red Giants: Distribution and Habitability

1. Introduction

In the distant future when our Sun becomes a Red Giant (RG), the habitable zone (HZ) in the Solar System may move towards the outer planets where the moons of Jupiter and Saturn might be candidates for our future generations to live (Sparrman 2022). Near-term considerations also prompt interest in exoplanet and exomoon systems of RG hosts as some of these worlds may presently be in the HZ of their parent star. In this paper we examine data from the NASA Exoplanet Archive, focusing on exoplanets around Red Giant (or Subgiant) stars.

When a star leaves the Main Sequence (MS) and begins its evolution into the Red Giant Branch (RGB), it undergoes a series of changes. As the fusion of hydrogen progresses in the core of a Main Sequence star, its effective temperature and luminosity increase slowly over time. At the end of a star’s Main Sequence stage its core is composed of helium while hydrogen begins to burn in the shell surrounding the core. The star then moves along the RGB of the Hertzsprung-Russell (H-R) diagram, with its temperature moderately decreasing, and its radius and luminosity significantly increasing.

As the host star evolves beyond MS, the orbits of its planets will also evolve. Due to the host’s mass loss, its surrounding planets will move outwards (Zahn 1977). On the other hand, tidal interactions tend to shrink the orbital radius of the planets (Villaver et al. 2014). In particular, Villaver et al. 2014 predicted that tidal interaction would cause planets to plunge into the star, and get engulfed, before a/R_s < 3, where a is the orbital semi-major axis of the planet and R_s is the stellar radius.

The first aim of our paper is to study the distribution of planets around RGs. Previous research (Jiang and Zhu 2018) found a power law relation between planet mass and stellar radius. The associated distribution of three variables is focused upon: the mass of the planet (M_p), the radius of the star (R_s), and the orbital semi-major axis (a). We aim to gain additional insight into the evolution of planets as the host star evolves post-Main Sequence.

To scientists, and the general public alike, habitability and the existence of extraterrestrial life is a topic of high interest (Kaltenegger 2017). A habitable zone is an annular region around a given star where any hosted planets have a relatively high likelihood of moderate average surface temperature, allowing for biological life (as we know it) to possibly exist. The HZ is usually determined primarily by the stellar energy flux from the host. More specifically, however, for a planet to be habitable it must not only be in its host’s HZ, but also possess the appropriate atmospheric and geological conditions accommodative to maintaining surface liquid water.

It has been predicted by many authors that as the Sun enters the RGB, Earth will no longer be in the Solar System’s HZ. As investigated by many studies (Danchi et al. 2005, Lopez et al. 2005, Cuntz et al. 2012, Ramirez and Kaltenegger 2016, Gallet et al. 2019, Sparrman 2022), post-MS evolution of the Sun will alter its HZ, possibly rendering some of the outer planets’ moons habitable to life such as found on Earth.

For a grid of stars with various mass and metallicity, Ramirez and Kaltenegger (2016) explored the evolution models of planets with their stars, and subsequent durations of planets in the HZ in detail. Their findings suggest three candidate systems that will become habitable once the host star becomes a RG. In this paper we apply the criterion used by two previous studies (Ramirez and Kaltenegger 2016, Sparrman 2022), initially proposed by Kopparapu and colleagues (Kopparapu et al. 2013), to current data in the NASA Exoplanet Archive (NEA), identifying those exoplanets in the HZ and discussing further parameterized regions not yet observed which may also contain habitable planets.

2. Data Collection and Distribution of Planets Around Red Giants

In this section, we briefly introduce our data collection and then discuss the distribution of Sub Giant (SG) and RG planets in the (M_p, a, R_s) parameter space. In Figure 1, we plot an H-R diagram of host stars using luminosity relative to the Sun ($L/L_{\odot }$) and stellar surface effective temperature (T_eff) values from the NEA, identifying 210 RG planets and 229 SG planets, as indicated by red × and blue +, respectively. Parameters of each planet and their host, namely (M_s, R_s, a, M_p) have been listed in Table A1 and Table A2 in the Appendix. To estimate the evolutionary stages of each star, we use MIST v1.2 stellar evolution tracks for solar metallicity stars with v/v_crit=0.4 (Dotter et al. 2016). In the figure, the purple curve indicates the end of the main sequence, while the separation between SG and RG can be seen from the shapes of the tracks. In this case, we have approximately separated SG and RG using black dashed lines. In comparison with Ghezzi et al. 2010, who used a color-independent M_bol=2.82 as the boundary between SG and RGs, our criterion further includes a few more stars, which, according to the evolutionary tracks, are in the RGB despite their relatively low luminosities. We caution that our identification of SG from MS stars can be inaccurate since we have not individually accounted for the metallicity of each host star.

We note that some of these planets, including 42 Dra, γ Dra (Döllinger and Hartmann 2021, henceforth referred to as D&H), and α Tau (Reicher et al. 2019), have been questioned as false positives. D&H further speculated that a substantial fraction of planets around K-giants with radii greater than 21R_⊙ can be false positives, based on the congregation of their orbital periods, lack of planet-metallicity correlation, as well as the excess number of planets around K-giants compared with MS stars. We shall make comparisons with D&H in Section 2.2 below.

2.1. Observed Evolution of Exoplanet Population as the Host Star Evolves

A previous study (Jiang and Zhu 2018) derived a planet mass-stellar radus relation for 150 exoplanets orbiting Red Giants:

$$M_p/M_{\oplus }=a{\left(R/R_{\odot }\right)}^b$$

(1)

with best-fit parameters a=150 and b=0.88. With the new data points we still see that there is a trend between M_p and R_s, yet their distribution appears to more be in a triangular distribution rather than a linear dependence. Further investigation of the origin of the M_p vs. R_s relation notes that the stellar radius tracks with the post-MS evolution stage of the host star. The fact that M_p increases with R_s corresponds to a relative lack of less massive planets around more evolved stars. In this paper, we shall use Figure 3 and Figure 4, in addition to Figure 2, to further investigate the evolution of the population of exoplanets around stars as they evolve.

In Figure 3a and 3b, we split R_s into three different intervals and plot MS/SG (silver dots) and RG planets in each interval separately as planet mass M_p (in Earth masses) vs. orbital semi-major axis a (in astronomical units). In particular, we separate RG planets into three categories according to R_s: R_s/R_⊙ < 5 (blue dots), 5 < R_s/R_⊙ < 25 (green dots), and R_s/R_⊙ > 25 (red dots). The (a, M_p) region occupied by RG planets shrinks as R_s increases — from its left side, with small a, from the bottom side, with low M_p, and from the right side, with large a. This shrinkage is best viewed from the right (b) panel of Figure 3, which focuses on the specific region of RG planets and adds contours generated via Kernel Density Estimate (KDE) for clarity. In Figure 4, we plot the orbital semi-major axis vs. stellar radius ratioed to solar radii.

At this stage, it is useful to point out the relation between Figure 2, Figure 3, and Figure 4. In Figure 3b, we separate the evolution of R_s into three bins (stages) and illustrate the lumped joint (a, M_p) distribution in each bin (stage). Figure 2 and Figure 4 each separately illustrates the continuous evolutions of the marginal distributions of M_p and a, respectively, as R_s increases, The step-wise shrinkage of (a, M_p) distribution as we progress from blue to green and to red, are continuously represented in Figure 4 for a and in Figure 2 for M_p.

2.2. Interpretations of the Evolutions in Population

In the following, we shall address the disappearance of planets with low semi-major axis (left side in Fig 3b), low mass (bottom side in Fig 3b), and high semi-major axis (right side in Fig 3b) separately.

For disappearance of planets with low semi-major axis, it is straightforward to anticipate planets with small orbital distance values to be engulfed and consumed as their host evolves and expands. According to Villaver et al. 2014, tidal interactions tend to speed up the engulfment of planets, and no planets should survive once a/R_s < 3. In Figure 4, we plot the orbital semi-major axis vs. stellar radius ratioed to solar radii, clearly illustrating that a/R_s = 3 is a cutoff and providing empirical evidence for tidally-accelerated engulfment. Correspondingly, in Figure 3b, we plot pink dashed lines to represent $a=15R_{\odot }$ and $a=75R_{\odot }$. These two lines indeed bound the green (5R_⊙<R_s<25R_⊙) and red (R_s>25R_⊙) populations from the left, respectively. As the stars evolve, this engulfment cutoff moves continuously along a.

Regarding the disappearance of low-mass planets with increasing R_s, we can see from Figure 3a that for stars with a radius less than 25R_⊙ many planets with masses 200 to 1000M_⊕ exist at distances 2 to 3 AU. Yet, such planets are not seen orbiting stars with R_s > 25R_⊙ --- even though much more massive planets are seen at the same distance. This disappearance of low-mass planets with increasing R_s corresponds directly from the M_p vs. R_s power-law fit obtained by Jiang and Zhu 2018. Note that Solar System planets lie on the lower part of the plot; only Jupiter is near the reach of current detection methods. However, Jovian mass exoplanets and comparable orbital distance (~5 AU) are not seen around Red Giants with R_s /R_⊙ > 25.

This disappearance can be explained using the limitations to the Radial Velocity (RV) method arising from the intrinsic oscillations of evolved stars (Hekker et al. 2008). Such oscillations have also been claimed to have led to false positives for exoplanets around RGs (Reichert et al. 2019). Hekker et al. 2008, who noticed that for stars with lower surface gravity g (i.e., larger radii), their measured minimum amplitudes of RV variations tend to increase, given approximately by

$$K_1^{\mathrm{int}}=2\times {10}^3{\left[g/\left({\mathrm{c}\mathrm{m}/\mathrm{s}}^2\right)\right]}^{-0.6}\mathrm{m}/\mathrm{s},$$

(3)

which they interpret as arising from intrinsic fluctuations of the star. Here, g is the surface gravitational acceleration of the star. For each RG, assuming e = 0, we obtain the minimum planet mass $M_{\mathrm{p}}^{\min }$ the star can host, this in order for the K₁ due to the planet to be greater than the intrinsic $K_1^{\mathrm{int}}$:

$$M_{\mathrm{p}}^{\min }=\sqrt{\frac{a{M}_s}{G}}{K}_1^{\mathrm{int}}.$$

(4)

In Figure 3b, we plot this minimum planet mass as a function of a, assuming $M_s=1M_{\odot }$ for $R=5R_{\odot }$ and $R={25}_{\odot }$ in purple dashed lines. These RV cutoffs approximately indicate the trend in which planets are cut off from the bottom, and to a lesser extent, the right.

We may further replace a in Equation (4) by its minimum value of $3R_s$ before engulfment, obtaining $M_{\mathrm{p}}^{\min }$ for each $R_s$, which is plotted as orange dots in Figure 2. In this plot, the RV cutoff indeed provides an excellent lower bound for the masses of planets detected using the RV method (red and blue dots) around substantially evolved stars.

Although the two cutoffs arise from different physical mechanism, given a particular population of planets, they do not act independently from each other; they have different efficiencies in cutting off populations depending on the distribution of planets in the a, M_p space. For example, since planets with smaller a tend to be low in M_p, they are less detectable by the RV method. Furthermore, for planets subject simultaneously to the engulfment and RV cutoffs, it is unclear whether they are actually engulfed or just unseen. Nevertheless, we would like to point out the region in Figure 3b bounded by the two pink lines and the upper purple line. The planets in this region represent a population that should be visible by the RV method, yet they are predicted to be engulfed by the $a=3R_{\odot }$ criterion. More specifically, we do see two planets with $R_s<25R_{\odot }$ host stars, yet the $R_s>25R_{\odot }$ population does not extend here. This provides some evidence that engulfment can indeed be taking place, and does contribute nontrivially to the shrinkage of the (a, M_p) distribution from the left. However, better detection methods not subject to the RV cutoff will be needed to more accurately study the engulfment phenomenon.

The disappearance of high semi-major axis planets as the star evolves cannot be fully explained with only the discussions above. As seen in Figure 3b, the red population has more concentrated values of a than simply applying the engulfment cutoff (the right pink line) and RV cutoff (the upper purple line) to the green and blue populations. Since orbital semi-major axis is highly correlated with orbital period due to the similarity in masses, our concentration in a is the same as the concentration of orbital periods (between 300 days and 800 days) for exoplanets around Red Giants with R_s > 21R_⊙, pointed out by D&H.

D&H argued that since the range of period falls within the period of intrinsic variations of stars (as modeled by Saio et al. 2015), and hence a fraction of these may not be actual planets. On the other hand, they provided plausible reasons for planets outside of this period range not to be discovered. For longer periods (corresponding to larger a), this could be due to the smaller RV variation being hidden under intrinsic fluctuations of the surface of the host star. For shorter periods (corresponding to smaller a), this could be due to the engulfment of planets by their host stars. Our discussions above quantitively explored these possibilities proposed by D&H. As we have seen, the engulfment and RV limitations do explain to some extent, but not completely, the concentration of periods described by D&H. In this way, our study lends support to D&H’s suggestion that some of these planets can be false alarms.

Disappearance of large a planets can also be explained from the inward migration of hosted planets, especially because large $R_s$ systems tend to be older, therefore the planets had more time to migrate. Finally, the fact that the more evolved host stars in our data tend to have lower metallicity and are older aged, and therefore were apt to have differently characterized populations of planets formed around them. However, such differences will likely have to be very substantial to be influential in this respect.

3. Habitable Planets Around Red Giants

In this section, we discuss the habitability of planets around RG and SG stars, briefly reviewing habitability criteria in 3.1, and presenting our findings in 3.2.

3.1. Criteria for Habitability

There exist multiple habitability conditions for a given exoplanet (or exomoon); most of which rely on the existence of water in liquid form to be present on at least a portion of that world’s surface. The simplest criterion uses equilibrium temperature, namely, the black-body radiation from the planet has to balance the radiation it absorbs from the star. If we define S as the flux of radiation from the host, this given by

$$S=\frac{L_s}{4\mathsf{\pi }{a}^2}$$

(5)

where $L_s$ is the star’s luminosity and $a$ is the orbital semi-major axis of the star’s exoplanet, the equilibrium temperature of the exoplanet is then given by

$$T_{\mathrm{eq}}=k{\left[\frac{S\left(1-A\right)}{4\mathsf{\sigma }}\right]}^{1/4}$$

(6)

where $A$ is the planetary albedo and $\mathsf{\sigma }$ is the Stefan-Boltzmann constant. The simplest habitability condition is $273\hspace{0.17em}\mathrm{K}<T_{\mathrm{eq}}<373\hspace{0.17em}\mathrm{K}$, with the low $T_{\mathrm{eq}}$ defining the Outer boundary of the Habitable Zone (OHZ) and the high T_eq defining the Inner boundary of the Habitable Zone (IHZ). The scalar quantity $k$ is a correction factor that can be used to approximately incorporate the greenhouse effect of an assumed planetary atmosphere, see Selsis et al. 2007. We adopt the Earth albedo of A=0.3 and use k=1.13, which reproduces the Earth surface temperature.

More realistic criteria exist in the literature. In this paper, we shall adopt two criteria obtained by previous study (Kopparapu et al. 2013) in which an effective solar flux is expressed in terms of

$$S_{\mathrm{eff}}\equiv S/S_{\oplus }$$

(7)

where $S_{\oplus }$ is the current solar energy flux at the location of the Earth, as well as the temperature T of the host star. Note that S_eff is dimensionless. In this paper, we shall adopt two different ways to define HZ boundaries, one conservative, the other optimistic. The conservative HZ accounts for greenhouse effects in the atmosphere of the planet, taking the inner boundary to be defined by the moist greenhouse effect where $S_{\mathrm{eff}}$ allows sufficient water vapor to exist in the stratosphere. The outer boundary is defined by the maximum heat retained by the planet while still providing habitable conditions. This is also known as the maximum greenhouse effect.

We also summarized the boundaries using the following fitting formula (Kopparapu et al. 2013) for the host star temperature range of 2600 K < T < 7200 K:

$$S_{\mathrm{limit}}\left(T\right)=S_0+a{T}_{*}+b{T}_{*}^2+c{T}_{*}^3+d{T}_{*}^4,T_{*}=T/K-5780\hspace{0.17em},$$

(8)

where values of a, b, c and d for conservative/optimistic, inner/outer boundaries are reproduced in Table 1. Parameters in equation (8) are all dimensionless. As noted in Table 1, a more optimistic approach uses the (theorized) history of Solar System planets Venus and Mars to determine the inner and outer bounds of the HZ. Here, the inner boundary of the HZ is based on the assertion that Venus has not had liquid water on its surface for only the past billion years – i.e., a billion years ago (recent) Venus might have had surface conditions suitable for water to exist. On the other hand, there is mounting evidence that (early) Mars had liquid water flowing on its surface 3.8 billion years ago. For these reasons, they define the inner boundary using the S_eff of recent Venus and the outer boundary using the S_eff of early Mars.

In Table 2, we list conservative and optimistic habitable zone RG-hosted planets. All planets are gas giants with masses ranging from 1 to 22 Jupiter masses (M_J). In the third and fourth columns, we list the spectral type and absolute V magnitude (obtained from apparent V-magnitude and distance data from NEA) of the host stars obtained from the NEA. Note that spectral type quoted here are not always consistent with other sources, and that classifications of stars using these values here may not be always consistent with our classification from the positions of the stars in the HR diagram (Fig 1). The hosts of planets in Table 2 are also shown as green dots in the H-R diagram of Figure 1. As can be readily perceived from Table 2 and Figure 1, these host stars are all in their early stages of evolution.

3.2. Red Giant Planets in the Habitable Zones

From the NASA Exoplanet Archive, we collected values for stellar luminosity and orbital semi-major axis to calculate S_eff. In Figure 5, we show the Red Giant planets on the T_eff vs. S_eff plot with lines indicating HZ boundaries. From the plot, it can be seen that there is a substantial difference between boundaries for the T_eq HZ and Kopparapu HZ. We highlighted Kopparapu et al. 2013 optimistically habitable planets in green.

Figure 6 shows Red Giant and Main Sequence planets on a semi-major axis vs. stellar radius plot with habitable planets indicated (light green dots for habitable planets around MS stars, and cyan dots for those around SGs, and dark green those around RGs). We also indicate, with purple line segments, the optimistic HZ of the host stars of all planets around RGs. As illustrated, habitable planets --- and indeed habitable zones --- tend to have increasing a as R_s increases, this is attributable to stars with larger radii --- and thus greater luminosity --- having HZs farther out. For detected HZ planets, they have a maximum R_s of ~ 8R_⊙, far below the maximum R_s of Red Giants, in this way, habitable planets so far discovered are either orbiting SG, or RG at their early stages of evolution.

As stars evolve beyond 8R_⊙, the HZ extends to larger a, beyond the region in which planets have been detected orbiting RGs. Around MS stars, planets do exist in this region, as is the case for the outer planets of the solar system (see Figure 3). Therefore, such planets might exist around RGs, even though they are not yet detectable.

For Main Sequence hosts, planets with the longest semi-major axes were all discovered by direct imaging (yellow dots in Figure 6). However, using the same detection method to find similar planets around RGs may be difficult due to the direct imaging method disfavoring systems with large contrast. More specifically, the contrast between a one solar-mass giant star and a potential planet is about 5 magnitudes larger that between a one-solar-mass main sequence star and a potential planet. In this way, the direct imaging method is not (yet) sufficient to detect planets around giant stars.

4. Conclusions and Discussions

In this paper we take new data from NASA’s Exoplanet Archive to update and further investigate trends regarding Red Giant systems. First, we revisit the Planet Mass-Stellar Radius relation previously found (Jiang and Zhu 2018). To further explore this trend, we separate Red Giant hosted exoplanets according to the radii of their hosts and plot planet mass against semi-major axis (Figure 3). As stellar radius increases, the region occupied by planets in the graph shrinks and for planets with smaller orbital semi-major axes, we found their disappearance to be consistent with tidal engulfment of planets where $a/R_s<3$ (Figure 4).

For the disappearance of planets with lower masses and those with larger orbital semi-major axes, this disappearance could be due to observational selection effects of the radial velocity method used to discover the vast majority of planets in these regions. Since lower mass and larger orbital semi-major axis correspond to lower amplitudes of radial velocity, the disappearance can be attributed to a higher detection threshold for the amplitude of radial velocity oscillations among more evolved Red Giants. We showed that in order for this selection effect to be the origin of such disappearance, the level of intrinsic RV fluctuation of Red Giants should depend on surface gravity following equation (3), which was proposed by Hekker et al. 2018. However, selection effects may also arise due to eccentricity and stellar mass. As this possibility is beyond the scope of this paper, further investigations of the origin of such selection effects are left to future studies.

Next, we examine the habitability of Red Giant exoplanets. To determine the habitable zone, we adopt criteria proposed by Kopparapu et al. 2013 and with this method found ten planets in the optimistic HZ, five of which are in the conservatively calculated HZ. However, all of these planets are gas giants and, therefore, very likely uninhabitable by life as we presently know it. Nevertheless, these planets may themselves host habitable exomoons. Even though a planet might be within the HZ at the moment, its total lifetime within the HZ may or may not be long enough for life to develop. As Ramirez and Kaltenegger 2016 have shown, depending on the planet-star configuration, a planet can stay for 0.2 – 9 Gyr in the post-MS HZ of a star, which does provide hope for life to develop on its moon. An obvious next step of research is to find out how long each of the planets in Table 2a and 2b had been in the HZ. For the solar system, on the other hand, Sparrman had shown that none of the outer planets will stay long enough in in the post-MS HZ of the sun for life to independently develop. Regardless of the time scale here, one can also speculate on the more exotic scenarios where life could have been hidden under the surface of the moons, or could move from nearby habitable planets to the outer planets as the host star evolves.

Finally, with habitable zone exoplanets identified, we revisited the issue of detection bias. We see that their orbital semi-major axis increases with stellar radii until $R_s/R_{\odot }$ ~ 8. However, this does not necessarily rule out further habitable zone exoplanets and it is very likely there are more HZ Red Giant exoplanets with a semi-major axis greater than ~ 4 AU. Even though some such planets can be seen around Main Sequence stars via direct imaging, similar planets around Red Giant stars have not yet been found. While the limitations of current imaging methods may preclude detecting planets around Red Giant stars, more advanced instrumentation coming online in the near term may enable this technique to be used for at least some Red Giant hosted exoplanetary systems. The next generation of space telescopes, such as the Habitable World Observatory (Gaudi et al. 2018), will have enhanced capabilities and will be able to observe smaller planets including those planets around Red Giant stars. In addition to static spectroscopy, it will be possible to observe variation in the reflected starlight spectra while the planet rotates around its axis. As in the case of the Earth, the surface of a rocky exoplanet is not expected to be homogeneous (if it has oceans, lands, forests, and deserts), nor is the cloud distribution (Li et al. 2022 and Jiang et al. 2018). These factors will have an impact on the time series of the exoplanetary spectrum.