Planetary Line-to-accretion Luminosity Scaling Relations: Extrapolating to Higher-order Hydrogen Lines

Planetary-mass companions have been detected or observed at emission lines (e.g., Haffert et al. 2019; Betti et al. 2022; Ringqvist et al. 2022; Wu et al. 2022). As for Classical T Tauri Stars (CTTSs), the lines trace the accretion process, and should originate from an accretion shock, magnetospheric accretion columns, or both (Aoyama et al. 2021; hereafter A021). Empirically, each line luminosity L_line correlates with total accretion luminosity L_acc, which is set mostly by the accretion rate $\dot{M}$. Thus, emission-line observations yield constraints on the formation mechanism and formation timescale of the accretors.

Different models attempt to convert emission-line luminosities into an accretion rate (Aoyama et al. 2018, 2020; Aoyama & Ikoma 2019; A021; Thanathibodee et al. 2019; Szulágyi & Ercolano 2020; Dong et al. 2021). A popular approach extrapolates the empirical L_acc–L_line correlation for CTTSs (e.g., Rigliaco et al. 2012; Alcalá et al. 2017; Komarova & Fischer 2020) down to planetary masses. In A021, we pointed out that at Hα luminosities L_Hα  ≲ 10⁻⁶ L_⊙ (near the L_Hα of PDS 70 b; Zhou et al. 2021), extrapolating Rigliaco et al. (2012) predicts more Hα emission than incoming energy. Thus, blind extrapolations might be invalid.

In A021 we gave L_line–L_acc scaling relations meant to apply at planetary masses. They are based on a simplified model of the accretion geometry but combined with detailed, NLTE hydrogen-line emission calculations (Aoyama et al. 2018). We computed and fit L_acc only up to the n = 8 energy level since the microphysical model follows electron populations only up to a certain level.

UVES observations of Delorme 1 (AB)b (Ringqvist et al. 2022) motivate us to extend the relationships to higher n, thinking also of CUBES (Alcalá et al. 2022) or NIR instruments. We fit the fit coefficients of our existing relations and extrapolate them to higher-order lines.

The correlation between L_acc and each L_line is usually written as (e.g., Alcalá et al. 2017)

$$\begin{eqnarray}&&{\mathrm{log}}_{10}\left({L}_{{\rm{acc}}}/{L}_{\odot }\right)=a\times {\mathrm{log}}_{10}\left({L}_{\mathrm{line}}/{L}_{\odot }\right)+b,\end{eqnarray} \tag{ 1 }$$

with a and b the fit coefficients. In A021, we considered a range of planet masses and accretion rates and fit the resulting L_line(L_acc). A hydrogen line is defined by the starting and final electron energy levels, n_i and n_f. Since the Aoyama et al. (2018) models go only up to n = 10, we did not study lines beyond n_i = 8 to avoid possible "edge effects." However, higher-level lines are in fact reliable enough (see Section 3). Therefore we generate L_line as in A021 for an ($\dot{M}$, mass) grid for H9, H10, Pa9, Pa10, and Br9, and fit Equation (1) to obtain their a and b.

Next, we plot the a and b of all lines as a function of n_i and n_f (Figure 1). We tried different functional forms, fitting the free parameters through gnuplot. We used only the n_f = 2–4 series, excluded the first (α) transition of each, and weighted by 1/(n_i − n_f). The goal was to have a reliable extrapolation beyond n ≈ 10 (from A021 or the lines added above). An excellent fit is:

$$\begin{eqnarray}&&a({n}_{{\rm{i}}},{n}_{{\rm{f}}})=0.811-\displaystyle \frac{1}{9.90{n}_{{\rm{f}}}-9.5{n}_{{\rm{i}}}}\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}\begin{array}{rcl}b({n}_{{\rm{i}}},{n}_{{\rm{f}}}) & = & \left[1+1.05\mathrm{ln}({n}_{{\rm{i}}})+\displaystyle \frac{1}{{n}_{{\rm{i}}}-{n}_{{\rm{f}}}}-\displaystyle \frac{1}{{n}_{{\rm{f}}}}\right]\\ & & \times (1.07+0.0694{n}_{{\rm{f}}})-1.41.\end{array}\end{eqnarray} \tag{ 2b }$$

Even though we excluded the α lines, their (a, b) predicted by Equation (2) agree closely with the ones from the direct fits.

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Thus, examples of planetary-regime scaling relations extending those of A021 are:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.83{\mathrm{log}}_{10}({L}_{{\rm{H}}9}/{L}_{\odot })+2.16\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{{\rm{H}}10}/{L}_{\odot })+2.27\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{{\rm{H}}11}/{L}_{\odot })+2.37\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{{\rm{H}}12}/{L}_{\odot })+2.47\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{{\rm{H}}13}/{L}_{\odot })+2.56\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{{\rm{H}}14}/{L}_{\odot })+2.64\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{{\rm{H}}15}/{L}_{\odot })+2.72\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{{\rm{H}}16}/{L}_{\odot })+2.80,\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.83{\mathrm{log}}_{10}({L}_{\mathrm{Pa}9}/{L}_{\odot })+2.60\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.83{\mathrm{log}}_{10}({L}_{\mathrm{Pa}10}/{L}_{\odot })+2.72\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{\mathrm{Pa}11}/{L}_{\odot })+2.82\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{\mathrm{Pa}12}/{L}_{\odot })+2.92\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{\mathrm{Pa}13}/{L}_{\odot })+3.01,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.83{\mathrm{log}}_{10}({L}_{\mathrm{Br}9}/{L}_{\odot })+2.98\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.83{\mathrm{log}}_{10}({L}_{\mathrm{Br}10}/{L}_{\odot })+3.08\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.83{\mathrm{log}}_{10}({L}_{\mathrm{Br}11}/{L}_{\odot })+3.19\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({L}_{{\rm{acc}}}/{L}_{\odot })=0.82{\mathrm{log}}_{10}({L}_{\mathrm{Br}12}/{L}_{\odot })+3.29,\end{eqnarray} \tag{ 19 }$$

simply evaluating Equation (2) for n_f = 2–4 and several n_i as examples. Where (a, b) from direct fits are available, the coefficients match excellently. Especially since the (a, b) capture only an average relation, Equation (2) can also be used for the Pfund or other series (n_f ≥ 5).

For error propagation, the errorbar on L_acc is σ ≈ 0.3 dex as for the other lines (Appendix A of A021). The species toolkit includes Equations (2)–(19) (Stolker et al. 2020; "Emission line" tutorial at https://species.readthedocs.io).

A few points deserve discussion:

• 1.  
  Why does it work?—The lines originate from the shock-heated gas below the shock. There, the cooling is slower than or comparable to the electron transitions from low-n levels. This makes non-equilibrium calculations necessary (Aoyama et al. 2018). However, contrary to the transitions from low-n levels that have a large energy difference, the transitions between high-n levels are faster than the cooling by more than orders of magnitudes. Thus, highly excited-hydrogen abundances follow the thermal (Boltzmann) distribution, which is monotonic, essentially exponential. At velocities and densities relevant here (Aoyama et al. 2020), electrons with n ≳ 7 are thermally equilibrated. Correspondingly, the line strength of transitions from those n is a simple function. The limiting behavior seen in Figure 1 reflects this.
• 2.  
  Domain of validity—All slopes in all series converge to a ≈ 0.82, whereas the intercepts are b ≈ 2–3, growing roughly logarithmically (Equation (2)). This increase of b reflects the intuitive result that a diminishing fraction of L_acc goes into lines of higher energy within a series (L_line/L_acc ∼ 10^−b since a ∼ 1). Formally, integrating L_line from n_f to infinity diverges, but actually Equation (2) will hold only below a maximum n_i. The highest observable n_i is likely smaller than this anyway.
• 3.  
  Compared to stars?—In the CTTS relations of Alcalá et al. (2017), a_⋆ ≈ 1.0–1.1 while b_⋆ ≈ 2.8–3.6, also increasing with n_i (Figure 1). At L_line values where the stellar and the planetary regimes somewhat overlap, this leads to a major, 1–4 dex discrepancy in L_acc between the two approaches A021. In the planetary-shock case, a much smaller fraction of the kinetic energy is converted into line luminosity.
• 4.  
  Physical context—The fits apply to shock emission, whether from magnetospheric or purely hydrodynamic accretion onto the surface of a planet or its CPD. However, no emission from accretion columns themselves is considered A021. Constraining their temperature structure observationally (Petrov et al. 2014) or theoretically would enable estimating their contribution.
• 5.  
  Which way forward?—The model of Aoyama et al. (2018) could be extended to calculate directly hydrogen lines involving n > 10 electrons, but for the purposes of L_acc–L_line relations our extrapolations should more than suffice. Adding He and metal lines would matter more. At preshock velocities v₀ ≳ 200 km s⁻¹, they and the hydrogen continuum carry a sizeable fraction of the energy (Aoyama et al. 2020), and have been detected (Eriksson et al. 2020; Zhou et al. 2021; Betti et al. 2022; Ringqvist et al. 2022).

In A021, we derived L_acc–L_line scalings for accreting planets. These scalings should replace uncalibrated extrapolations of the stellar relations down to low line luminosities, which can yield unphysical results. Here, we fit the fit coefficients for the Balmer, Paschen, and Brackett series and extrapolated them to higher-level lines. Thanks to the simplicity of the hydrogen atom, we argued that this estimates well the results of detailed calculations, also for higher series. The underlying scenario is a shock in which hydrogen lines carry most of the energy. Thus this extension is valid in the same domain of preshock conditions as the main model: for preshock number densities n₀ ∼ 10⁹–10¹⁴ cm⁻³ and preshock velocities up to v₀ ≈ 200 km s⁻¹ (Aoyama et al. 2020). Equation (2) can be used to generate scalings easily, with examples in Equations (3)–(19). These extended relations allow estimates of the accretion rate of planetary-mass objects from even more hydrogen lines than up to now.

G.-D.M. acknowledges support from the DFG priority program SPP 1992 "Exploring the Diversity of Extrasolar Planets" (MA 9185/1), and from the Swiss National Science Foundation (SNSF), grant 200021_204847 "PlanetsInTime." Y.A. acknowledges support from the National Key R&D Program of China (No. 2019YFA0405100). Parts of this work have been carried out within the framework of the NCCR PlanetS supported by the SNSF.