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arxiv: 2605.18950 · v1 · pith:2LRFZHIRnew · submitted 2026-05-18 · 🌌 astro-ph.EP

Planetary formation tracks on the Hertzsprung-Russell diagram: Visualising the processes of giant planet growth

Benedikt Gottstein , Gabriel-Dominique Marleau , Christoph Mordasini This is my paper

Pith reviewed 2026-05-20 07:43 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords planetary formationHertzsprung-Russell diagramgiant planet growthaccretion processesluminosity-temperature trackscooling tracksdirectly imaged planetsBern model
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The pith

Forming giant planets trace three distinct branches on an extended Hertzsprung-Russell diagram.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the Hertzsprung-Russell diagram to planets to visualize how solid and gas accretion shape their luminosity and temperature during growth. It finds an ascending track while solids dominate, a near-horizontal track after the planet detaches from the disk and gas accretion slows, and a descending cooling track once formation ends. These paths depend on details like the size of accreted solids, migration, and whether gas is accreted hot or cold. A reader would care because the diagram offers a new way to interpret data on young planets and connect formation theory to observations of directly imaged worlds.

Core claim

Planetary HRDs exhibit three branches corresponding to successive phases: an ascending branch during solid-dominated growth, strongly set by the size of accreted bodies and by migration, with an analytic result of L proportional to T to the eighth for in-situ planetesimal accretion; a near-horizontal branch beginning at detachment when gas accretion becomes disk-limited, where hot accretion, higher masses, and pebble accretion bend tracks upward; and a descending branch where accretion ends and planets join constant-mass cooling tracks with weak radius evolution and L similar to T to the fourth. The tracks are computed using the Bern model coupled to radiation-hydrodynamical simulations for

What carries the argument

The three-branch structure of planetary formation tracks on the Hertzsprung-Russell diagram, enabled by coupling the Bern planet formation model with radiation-hydrodynamical simulations to determine time-dependent shock heating efficiency.

If this is right

  • In-situ planetesimal accretion leads to an analytic scaling where luminosity is proportional to temperature to the eighth power on the ascending branch.
  • After detachment, electron degeneracy increases, lowering interior temperatures and stabilizing planetary radii.
  • Hot gas accretion and pebble accretion cause the tracks to bend upward on the near-horizontal branch.
  • Once accretion ceases, planets follow standard cooling tracks where luminosity scales roughly as temperature to the fourth power.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Future direct imaging of young embedded planets could test these tracks if models account for circumplanetary disk emission.
  • The framework might be applied to populations of lower-mass planets to see if similar branches appear.
  • Observational placement of planets on such diagrams could constrain the dominant mode of solid accretion in real systems.

Load-bearing premise

The Bern model combined with radiation-hydrodynamical simulations accurately captures time-dependent accretion-shock heating efficiency and interior structures throughout the entire formation and early evolution phases.

What would settle it

Detection of a young giant planet whose measured luminosity and temperature place it outside the three predicted branches or violate the L proportional to T to the eighth relation during the solid accretion phase.

Figures

Figures reproduced from arXiv: 2605.18950 by Benedikt Gottstein, Christoph Mordasini, Gabriel-Dominique Marleau.

Figure 2
Figure 2. Figure 2: Evolution of the relative shock heating k as a function of time during the gas accretion of the forming giant planet in the detached phase. The data shown is from the default case planet from Sect. 3.1. The mass and radius evolution can be seen by their values at selected data points. The inset highlights the brief hot phase after detachment. rolle et al. 1998; Thanathibodee et al. 2019) or boundary layer … view at source ↗
Figure 3
Figure 3. Figure 3: HR diagram track of the default case (final mass of 2.2 MJ ; see [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time evolution, up to 20 Myr, of selected quantities related to (a) luminosity, (b) temperature, (c) mass, (d) radius, (e) mass accretion, and (f) time scales. In each panel, a dashed (dot-dashed) line marks the moment of detachment (disk dispersal). Details: (a) Total luminosity emerging from the planet surface Ltot (without the fraction of the shock luminosity that is radiated away in the accretion shock… view at source ↗
Figure 5
Figure 5. Figure 5: The planetary horizontal branch up to the begin of the pure cooling (evolution) phase for the default, variable-η (green; as in [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Top panel: L–T track of a planet with forced total stop of accretion immediately after detaching with Mtot = 39.55 M⊕, Menv = 21.90 M⊕ and a radius of 83.11 RJ . The point of detachment and end of accretion is marked with a star. The point with maximal Ts, core = 23 500 K in the forced evolution stage is marked with a trian￾gle, while the point with Ts, max = 436 K is marked with a dot. Bottom panel: Z¯ an… view at source ↗
Figure 7
Figure 7. Figure 7: HRD tracks for planets with a range of final masses. The different masses are obtained by changing the initial disk gas mass at a fixed solid surface density Σpla,5.2 au = 10 g cm−2 . The initial disk masses of 0.025, 0.03, 0.035, 0.04, 0.05, 0.075 and 0.1 M⊙ lead to the final masses colour￾coded in the figure in the same increasing order. Stars indicate the moment of detachment, while dashed lines connect… view at source ↗
Figure 8
Figure 8. Figure 8: Top left panel: HRD for a formation scenario where the solid core grows via pebble accretion. Initial gas disk masses are 0.02 (blue), 0.03 (orange), 0.04 (green) and 0.05 M⊙ (red). The planets reach a total mass of respectively 1.4, 2.8, 4.6 and 6.7 MJ , all with similar core masses of 8–10 M⊕. We show data up to 200 Myr. Other panels: Time evolution of the luminosity (without the accretion shock luminosi… view at source ↗
Figure 9
Figure 9. Figure 9: L–T tracks (top panel) of three migrating planets with initial semi-major axis of 10 au (blue), 12 au (orange) and 14 au (green) showing their formation and evolution up until 200 Myr. Filled dots and squares show the point of detachment and disk dispersal, respectively. Labels (a)–(h) mark various phases in the process, explained in the top right. Type I (Type II) migration is indicated by dashed (solid) … view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of the semi-major axis, luminosity, radius and total mass for three migrating planets with different initial starting location of 10 au (blue), 12 au (orange) and 14 au (green). The point of detachment is marked by a solid dot and the point of disk dispersal by a solid square. Data is shown for the formation phases and until 200 Myr of the evolution phase and not for the full duration as previou… view at source ↗
Figure 11
Figure 11. Figure 11: Left panel: L–T tracks for migrating planets from [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
read the original abstract

The Hertzsprung-Russell diagram (HRD) is central to stellar astrophysics but has rarely been used to interpret planet formation. We extend the HRD concept to forming planets and study how solid and gas accretion, cooling/contraction, and migration shape luminosity-temperature tracks in different formation scenarios. We compute planetary interior structures throughout formation and evolution with the Bern model and, for the first time, couple it to radiation-hydrodynamical simulations to obtain a time-dependent accretion-shock heating efficiency, helping to address the cold-/hot-start ambiguity. Planetary HRDs exhibit three branches corresponding to successive phases: (i) an ascending branch during solid-dominated growth, strongly set by the size of accreted bodies (and thus the solid accretion rate) and by migration; for in-situ planetesimal accretion we find analytically $L \propto T^8$. (ii) A near-horizontal branch beginning at detachment when gas accretion becomes disk-limited and contraction accelerates; hot accretion, higher masses, and pebble accretion bend tracks upward. Increasing electron degeneracy after detachment lowers interior temperatures and stabilises radii. (iii) A descending branch where accretion ends and planets join constant-mass cooling tracks with weak radius evolution and $L \sim T^4$. Our tracks agree well with synthetic populations and are broadly consistent with directly imaged planets. Populating the short-lived early branches observationally will be difficult, and embedded accreting planets require models including accretion-shock emission and circumplanetary-disk reprocessing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends the Hertzsprung-Russell diagram concept to forming giant planets by computing luminosity-temperature tracks with the Bern 1D formation model. For the first time, this model is coupled to radiation-hydrodynamical simulations to obtain a time-dependent accretion-shock heating efficiency. The resulting tracks display three branches: (i) an ascending branch during solid-dominated growth (with an analytical L ∝ T^8 relation derived for in-situ planetesimal accretion, controlled by accreted body size and migration), (ii) a near-horizontal branch after detachment when gas accretion becomes disk-limited, and (iii) a descending branch on constant-mass cooling tracks with L ∼ T^4. The tracks are compared to synthetic populations and directly imaged planets; the work notes observational challenges for early phases and the need for circumplanetary-disk reprocessing in future models.

Significance. If the central results hold, the paper supplies a useful visualization framework that links specific formation processes (solid accretion rate, migration, hot vs. cold accretion, degeneracy effects) to observable HRD loci. The analytical L ∝ T^8 scaling and the explicit coupling to radiation-hydrodynamical runs to address the cold-/hot-start ambiguity are concrete strengths that could help interpret future direct-imaging data of young planets and guide population synthesis comparisons.

major comments (2)
  1. [Methods section on model coupling] The coupling between the Bern 1D model and the radiation-hydrodynamical simulations is described at a high level only (abstract and methods). No explicit statement is given on how entropy or luminosity continuity is enforced at the shock boundary, nor on how circumplanetary-disk reprocessing (explicitly flagged as still required) is omitted. Because the ascending-branch slope and the location of detachment both depend directly on the adopted shock-heating efficiency, this interface detail is load-bearing for the claimed three-branch structure.
  2. [Section presenting the analytical relation] The analytical derivation of L ∝ T^8 for in-situ planetesimal accretion is stated without the intermediate steps, the assumed solid accretion rate scaling, or the interior temperature profile used. This relation underpins the physical interpretation of the ascending branch and its dependence on body size; its robustness cannot be assessed from the given information.
minor comments (2)
  1. [Figure captions] Figure captions should explicitly label the three branches, indicate which parameters are varied in each panel (e.g., planetesimal vs. pebble accretion, migration rate), and note the assumed accretion-shock efficiency.
  2. [Abstract and results] The abstract claims broad consistency with directly imaged planets but does not quantify the comparison (e.g., overlap in luminosity or temperature ranges); a short table or inset would strengthen this statement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive report and positive assessment of the work. Their comments identify areas where additional detail will improve clarity. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [Methods section on model coupling] The coupling between the Bern 1D model and the radiation-hydrodynamical simulations is described at a high level only (abstract and methods). No explicit statement is given on how entropy or luminosity continuity is enforced at the shock boundary, nor on how circumplanetary-disk reprocessing (explicitly flagged as still required) is omitted. Because the ascending-branch slope and the location of detachment both depend directly on the adopted shock-heating efficiency, this interface detail is load-bearing for the claimed three-branch structure.

    Authors: We agree that the coupling requires more explicit description. In the revised manuscript we have added a dedicated subsection in Methods that details the interface: luminosity continuity is enforced by matching the interior luminosity to the time-dependent shock luminosity computed from the RHD runs, while entropy continuity is achieved by adopting the post-shock entropy from the simulations as the outer boundary condition for the 1D structure integration. Circumplanetary-disk reprocessing is omitted, as already noted in the abstract; the present study isolates the effect of the accretion-shock efficiency, with CPD reprocessing identified as a necessary future extension. These additions directly address the dependence of the ascending branch and detachment on the adopted heating efficiency. revision: yes

  2. Referee: [Section presenting the analytical relation] The analytical derivation of L ∝ T^8 for in-situ planetesimal accretion is stated without the intermediate steps, the assumed solid accretion rate scaling, or the interior temperature profile used. This relation underpins the physical interpretation of the ascending branch and its dependence on body size; its robustness cannot be assessed from the given information.

    Authors: We thank the referee for this observation. The L ∝ T^8 scaling was derived under the assumptions of in-situ planetesimal accretion with solid accretion rate scaling as Ṁ_solid ∝ R_p² and an interior temperature profile obtained from radiative-convective equilibrium with constant opacity. We have now included the full step-by-step derivation, with all intermediate equations and assumptions, as Appendix A of the revised manuscript. This makes the physical origin of the relation and its sensitivity to accreted body size fully transparent and reproducible. revision: yes

Circularity Check

0 steps flagged

No circularity detected; tracks are model outputs, not self-definitions

full rationale

The paper generates planetary HRD tracks by running the Bern model coupled to external radiation-hydrodynamical simulations that supply time-dependent accretion-shock heating efficiencies. The three-branch structure, the detachment point, and the stated analytic scaling L ∝ T^8 for in-situ planetesimal accretion are presented as direct consequences of those computations and of the phase-dependent accretion regimes, rather than as quantities fitted to or defined by the target result itself. No self-citation chain is invoked to justify uniqueness, no ansatz is smuggled via prior work, and no fitted parameter is relabeled as a prediction. The derivation therefore remains independent of the final tracks and rests on the external simulations and the model's internal equations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claims rest on the accuracy of the Bern planet formation model for interior structures and on the validity of coupling it to radiation-hydrodynamical simulations for time-dependent shock heating; no explicit free parameters or invented entities are named in the abstract.

free parameters (1)
  • accretion-shock heating efficiency
    Described as time-dependent and obtained from radiation-hydrodynamical simulations; exact functional form or fitting details not provided in abstract.
axioms (1)
  • domain assumption The Bern model correctly computes planetary interior structures and evolution throughout formation
    Invoked to generate the tracks for solid and gas accretion, cooling, contraction, and migration phases.

pith-pipeline@v0.9.0 · 5809 in / 1327 out tokens · 84608 ms · 2026-05-20T07:43:22.807487+00:00 · methodology

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