Planetary line-to-accretion luminosity scaling relations: Extrapolating to higher-order hydrogen lines

Aoyama et al. (2021, ApJL) provided scaling relations between hydrogen-line luminosities and the accretion luminosity for planetary-mass objects. These fits should be an improvement over blind extrapolations of stellar relations. The fits go up only to the n = 8 electron energy level, but higher-n Balmer lines have been observed in the near-UV at Delorme 1 (AB)b with UVES (Ringqvist et al. 2023). We extend the scaling relations to higher-n levels for the Balmer and other series by fitting the fit coefficients (a, b) themselves and extrapolating them. Within the assumption of an accretion shock as the source of line emission, these fits should be robust for accreting planetary-mass objects.


MOTIVATION
Planetary-mass companions have been detected or observed at emission lines (e.g., Haffert et al. 2019;Wu et al. 2022;Betti et al. 2022a,b;Ringqvist et al. 2023). 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 Ao21). Empirically, each line luminosity L line correlates with total accretion luminosity L acc , which is set mostly by the accretion rateṀ. 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(Aoyama et al. , 2020Aoyama & Ikoma 2019;Ao21;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 Ao21, we pointed out that at H α luminosities L H α 10 −6 L (near the L H α of PDS 70 b; Zhou et al. 2021), extrapolating Rigliaco et al. (2012 predicts more H α emission than incoming energy. Thus, blind extrapolations might be invalid. gabriel.marleau@uni-tuebingen. de, yaoyama@pku.edu.cn In Ao21 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. 2023) 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.

FIT OF THE FIT COEFFICIENTS, AND THEIR EXTRAPOLATION
The correlation between L acc and each L line is usually written as (e.g., Alcalá et al. 2017) log 10 (L acc /L ) = a × log 10 (L line /L ) + b, with a and b the fit coefficients. In Ao21, 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 §3). Therefore we generate L line as in Ao21 for an (Ṁ, mass) grid for H9, H10, Pa9, Pa10, and Br9, and fit Eq. (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 (Fig. 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 Ao21 or the lines added above). An excellent fit is: Even though we excluded the α lines, their (a, b) predicted by Eq. (2) log 10 (L acc /L ) = 0.82 log 10 (L Pa13 /L ) + 3.01, simply evaluating Eq.
(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, Eq.

DISCUSSION
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 behaviour seen in Fig. 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 (Eq. (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 Eq.
(2) will hold only below a maximum n i . The highest observable n i is likely small 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 (Fig. 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 (Ao21). 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 (Ao21). 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 v 0 200 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. 2022a,b;Ringqvist et al. 2023).

SUMMARY
In Ao21, 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 n 0 ∼ 10 9 -10 14 cm −3 and preshock velocities up to v 0 ≈ 200 km s −1 (Aoyama et al. 2020). Eq. (2) can be used to generate scalings easily, with examples in Eqs. (3)-(19). These extended relations allow estimates of the accretion rate of planetary-mass objects from even more hydrogen lines than up to now.