Reproducing the stellar-mass dependence of the giant planet occurrence rate with pebble accretion models

This study utilizes a pebble-driven core accretion population synthesis model, constrained by homogenized radial velocity survey data, to demonstrate that assuming higher accretion rates and shorter disk lifetimes around more massive stars successfully reproduces the observed peak in giant planet occurrence around 1.7–2 solar mass stars.

The Gist

Imagine the universe as a giant cosmic bakery. In this bakery, stars are the ovens, and the swirling disks of gas and dust around them are the dough. The bakers' goal? To bake giant planets (like Jupiter) out of this dough.

For a long time, astronomers noticed a strange pattern in this bakery: Giant planets seem to love ovens of a specific size. They are rare in small ovens (low-mass stars), incredibly common in medium-sized ovens (stars about 1.7 times the mass of our Sun), and then they vanish almost entirely in the massive, roaring ovens (stars more than 2.5 times the mass of our Sun).

Why does this happen? Why do the "sweet spots" for giant planets exist only in the middle?

In this paper, Heather Johnston and her team decided to run a cosmic simulation to figure it out. They built a virtual bakery using a model called "pebble accretion." Here is how they did it, explained simply:

1. The Recipe: Pebbles and Gas

Think of building a giant planet like building a sandcastle.

• The Pebbles: Tiny rocks and ice grains (pebbles) drift through the disk.
• The Core: A baby planet (embryo) starts as a small lump of sand. It grows by scooping up these pebbles.
• The Gas: Once the sandcastle gets big enough, it starts sucking in the surrounding air (gas) rapidly, puffing up into a giant planet.

The team simulated this process for thousands of different stars, changing the size of the star and the "ingredients" (metallicity) in the dough.

2. The Secret Ingredient: The Accretion Rate

The key discovery in this paper is about how fast the dough is being eaten by the oven.

• The Problem with Small Ovens (Low-Mass Stars): In smaller stars, the "dough" (gas and dust) falls onto the star very slowly. It's like a slow drip. The baby planets have plenty of time to grow, but they are starving because the pebbles aren't arriving fast enough. They never get big enough to start sucking in the gas. Result: Few giant planets.
• The Problem with Massive Ovens (High-Mass Stars): In huge, hot stars, the dough is being devoured incredibly fast. The "oven" is so hot and bright that it blows the dough away (evaporates the disk) before the baby planets can finish building their sandcastles. The disk disappears before the planets can grow up. Result: Few giant planets.
• The Sweet Spot (The 1.7 Solar Mass Star): This is the Goldilocks zone. The star is massive enough to have a lot of ingredients (pebbles) arriving quickly, but not so massive that it blows the whole bakery away before the planets are born. The baby planets can grow fast enough to grab the gas before the disk vanishes. Result: A peak in giant planet numbers.

3. The "Fast-Forward" Effect

The team also found something fascinating about where and when these planets are born around massive stars.

Imagine a race.

• Around small stars, the race is slow. The planets have to stay close to home (near the star) to gather enough pebbles over a long time.
• Around massive stars, the race is a sprint. Because the disk is disappearing so fast, the baby planets have to grow super quickly. To do this, they need to be born further out in the disk where there is more raw material. They grow huge, very fast, and far away from the star.

The Catch: Even though they are born far away (10–20 AU), they usually migrate inward later, ending up close to the star where we can see them. But if they are born too far out around a massive star, they might not make the trip inward before the disk disappears, leaving them hidden in the dark. This explains why we see almost no giant planets around the biggest stars.

4. The Conclusion: It's All About Timing

The main takeaway is that the rate at which the star eats its own disk is the most important factor.

• If the star eats too slowly, the planets starve.
• If the star eats too fast, the planets are swept away before they can finish.
• If the star eats at just the right speed (around 1.7 solar masses), the planets get the perfect amount of food at the perfect time to become giants.

Why This Matters

This study helps us understand why our own Solar System (around a medium-sized star) has Jupiter, but why we might not find many giant planets around the biggest, brightest stars in the galaxy. It also suggests that if we want to find more giant planets around massive stars, we might need to look further out from the star, or use different telescopes, because they might be hiding in the outer regions where they were born.

In short: Giant planets are like delicate soufflés. You need the right amount of heat and the right amount of time. Too little heat, they don't rise. Too much heat, they burn. The universe has found the perfect oven setting for giant planets, and it's right in the middle.

Technical Summary

Technical Summary: Reproducing the Stellar-Mass Dependence of the Giant Planet Occurrence Rate with Pebble Accretion Models

1. Problem Statement
Observational surveys (e.g., Reffert et al. 2015; Wolthoff et al. 2022) have established that the occurrence rate of giant planets ($\eta_J$) is not uniform across stellar masses. Instead, $\eta_J$ rises with stellar mass up to a peak around $1.7\text{--}2.0,M_\odot$and then declines sharply for stars more massive than$2.5,M_\odot$, where giant planets are virtually absent. The physical mechanisms driving this specific non-monotonic dependence remain unclear. While metallicity dependence is well-understood, the interplay between stellar mass, disk evolution, and planet formation efficiency requires further theoretical constraint.

2. Methodology
The authors conducted a population synthesis study using a pebble-driven core accretion model (based on Liu et al. 2019; Johnston et al. 2024) to simulate giant planet formation across a range of stellar masses ($0.9\text{--}5,M_\odot$) and metallicities.

• Stellar Sample: The models were constrained using a homogenized sample of 482 stars from three radial velocity (RV) surveys (EXPRESS, PPPS, and Lick giant star survey) compiled by Wolthoff et al. (2022).
• Disk Physics:
  • Viscosity & Evolution: Adopted an $\alpha$-viscosity prescription ($\alpha_g = 10^{-2}$) with disk accretion rates evolving via viscous accretion and X-ray driven photoevaporation.
  • Stellar Luminosity: Incorporated evolving pre-main sequence (PMS) stellar luminosity tracks (Siess et al. 2000). Crucially, this accounts for the fact that massive stars ($M_* > 1.5\,M_\odot$) increase in luminosity as they approach the main sequence, heating their disks and increasing gas scale heights.
  • Structure: Modeled a two-component disk (inner viscously heated, outer irradiated) with a transition radius. Pebbles were assumed to be a mix of water ice and silicates.
• Planet Formation:
  • Core Growth: Embryos formed via streaming instability. Core growth proceeded via pebble accretion until reaching the pebble isolation mass ($M_{iso}$), at which point solid accretion halted.
  • Gas Accretion: Runaway gas accretion was triggered once the core exceeded $M_{iso}$.
  • Migration: A combined Type I/II migration formula was used.
• Experimental Design: For each stellar mass and metallicity combination in the observational sample, 100 Monte Carlo simulations were run. The study tested three different prescriptions for the initial mass accretion rate ($\dot{M}_{acc}$) dependence on stellar mass:
  1. M17: Based on low-mass T Tauri stars (Manara et al. 2017).
  2. W20: Based on intermediate-mass Herbig Ae/Be stars (Wichittanakom et al. 2020).
  3. J24: A new empirical estimate by the authors to correct for age biases in the W20 sample (accounting for the fact that observed accretion rates in older massive stars may be underestimated as initial conditions).

3. Key Contributions

• Identification of the Critical Parameter: The study demonstrates that the stellar-mass dependent accretion rate is the primary driver of the observed giant planet occurrence peak.
• Correction of Age Bias: The authors highlight that using raw observational accretion rates for massive stars without correcting for stellar age leads to incorrect model predictions. They propose a corrected relation (J24) that better reflects initial disk conditions.
• Mechanism for the "Drop-off": The paper elucidates why giant planets are rare around stars $>2.5\,M_\odot$: high accretion rates in these systems lead to rapid disk dispersal (short lifetimes), preventing embryos from accumulating sufficient mass before the gas is removed.
• Migration vs. Formation Location: The study distinguishes between the formation location (where runaway gas accretion occurs) and the observed location, showing that significant inward migration is required to reconcile the two.

4. Results

• Occurrence Rate Reproduction:
  • The M17 model (low accretion rates for massive stars) failed, predicting a monotonic increase in planet occurrence with mass, contradicting observations.
  • The W20 model (high accretion rates) reproduced the peak but overestimated the occurrence rate for stars $>2.5\,M_\odot$ and showed a too-shallow decline.
  • The J24 model (corrected high accretion rates) successfully reproduced the observed distribution: a peak at $\sim1.7\,M_\odot$ and a sharp decline for $M_* > 2.5\,M_\odot$.
• Formation Timescales and Locations:
  • Time: Giant planets form significantly earlier around massive stars (e.g., $\sim0.5$ Myr for $4,M_\odot$vs.$\sim1$Myr for$1,M_\odot$) due to higher accretion rates.
  • Location: Runaway gas accretion occurs at larger orbital radii for more massive stars (up to $>10$ AU for $M_* > 2.5\,M_\odot$) compared to low-mass stars ($\lesssim5$ AU). This is because the rapidly dispersing disks of massive stars force planets to accrete gas further out before the disk dissipates.
• Metallicity Dependence: Both models confirmed a positive correlation between metallicity and planet occurrence, as higher metallicity enhances pebble flux and core growth rates.
• Discrepancy Resolution: The large gap between simulated formation radii ($5\text{--}20$AU) and observed RV detections ($\lesssim5$ AU) is explained by substantial inward migration occurring after the runaway gas accretion phase.

5. Significance

• Theoretical Validation: This work provides strong evidence that pebble accretion, when coupled with realistic, stellar-mass-dependent disk evolution, can naturally reproduce the complex, non-monotonic distribution of giant planets observed in the galaxy.
• Disk Lifetime Constraints: It reinforces the hypothesis that disk lifetime is the limiting factor for giant planet formation around massive stars. The "Goldilocks" zone for giant planet formation exists where accretion rates are high enough to build cores quickly but low enough to allow the disk to survive long enough for gas accretion.
• Future Observational Targets: The study predicts a population of giant planets on wide orbits ($5\text{--}15$ AU) around intermediate-mass stars that have not yet migrated inward. These are currently difficult to detect via RV or direct imaging but are prime targets for astrometric surveys (e.g., Gaia and future missions).
• Atmospheric Implications: The formation of giant planets at large radii around massive stars implies accretion of gas rich in CO but poor in H$_2$O (due to ice lines), potentially leading to super-stellar C/O ratios in their atmospheres.

In conclusion, the paper successfully bridges the gap between theoretical planet formation models and observational statistics by identifying the accretion rate as the critical variable governing the stellar mass dependence of giant planet occurrence.