Semianalytical Accretion-Tracer Emission: Forming Planets Are Intrinsically Faint

TL;DR

The study develops a semianalytical, 2.5D framework for predicting hydrogen-line emission from forming gas giants, focusing on ballistic infall from the Hill sphere and excluding magnetospheric accretion. By coupling an extended Ulrich-type flow with shock-emission models and calibrating to population-synthesis statistics, the authors show that only a small fraction of Hill-sphere inflow produces detectable line emission, yielding maximum Hα luminosities around $L_{ ext{line,max}}\approx2\times10^{-7}\,L_\odot$ that are roughly mass-independent. Compared with CTTS-based extrapolations, the predicted line luminosities are 3–4 orders of magnitude fainter, explaining the paucity of detections and suggesting that many forming planets could remain hidden even in deep surveys. The work provides a practical tool for interpreting non-detections, constraining CPD transport timescales, and guiding future high-sensitivity, close-in searches for forming planets, while noting that magnetospheric accretion could raise line fluxes in some cases. Overall, the results imply that accreting planets are intrinsically faint tracers and that closer-in, deeper observations are needed to reveal the population of forming super-Jupiters.

Abstract

Direct-imaging surveys have looked for accreting planets through their accretion tracers such as H alpha but have been less fruitful than expected. However, up to now, hydrogen-line emission at accreting planets has been estimated primarily with extrapolations of stellar-scaling relationships or with theoretical spherically-symmetric computations. To predict the line emission intensity during the formation phase, we follow the consequences of angular momentum conservation of the material accreting onto a gas giant in a protoplanetary disc. We focus on the limiting case that magnetospheric accretion does not occur, which yields a conservative estimate of the line emission and should correspond to certain epochs during formation. We extend but simplify an existing analytical description of the multidimensional gas flow onto an accreting gas giant, the ballistic infall model, and combine this with detailed shock emission models. Applying this to data from a global planet formation model, we confirm that the line-emitting accretion rate is a minuscule fraction of the gas inflow into the Hill sphere. Also, forming planets are mostly fainter than PDS 70 b and c or WISPIT 2 b, with a maximum H alpha line luminosity Lline near 1e-7 Lsol, roughly independent of planet mass. Most surveys have not been sensitive to such faint planets. Other hydrogen lines in the NIR are fainter by 1--2 dex. This implies that accreting planets are fainter than from past estimates, such that the non-detections are not as constraining as thought. A deeper look closer in to the host stars could well reveal many forming super-Jupiters.

Figures (13)

• Figure 1: Flow pattern in the poloidal plane ("side view") for the fiducial parameters in m22Schock. We set ${f_{\textrm{cent}}}\xspace=0.03$ (see Figure \ref{['Abb:flowducialvarfzent']} for ${f_{\textrm{cent}}}\xspace=1/3$). For illustrative purposes, the CPD (blue region) ends at $r=\xi{R_{\textrm{Hill}}}\xspace$ with $\xi=0.4$martin23, with the region inside ${R_{\textrm{cent}}}\xspace={f_{\textrm{cent}}}\xspace{R_{\textrm{Hill}}}\xspace$ highlighted (dark blue). We plot implicitly the exact solution of the orbit equation (solid lines) and the small-$\theta_0$ solution (dashed red lines; pale for $\theta_0>30{\hbox{\textdegree}}\xspace$). The green-yellow line is the $\theta_0=20{\hbox{\textdegree}}\xspace$ streamline from m22Schock, which is close to the $\theta_0=15{\hbox{\textdegree}}\xspace$ streamline at our $r_0={R_{\textrm{Hill}}}\xspace$. Streamlines starting closer to the midplane are shown thicker to remind of the density stratification (Equation \ref{['Gl:rhor0']}). The grey arc near ${R_{\textrm{Hill}}}$ displays the height of the CPD ($90{\hbox{\textdegree}}\xspace-\theta_{\textrm{CPD}}\xspace$) and we will assume that there is in fact no inflow over those $\theta_0$ (Sections \ref{['Th:RBdusse']} and \ref{['Th:MPkt']}).
• Figure 2: Inflow functions (that is, mass loading at the Hill sphere) as in taylor24 (coloured lines) and with an example of our adaptation for massive accreting planets ("Gaussian"; ${R_{\textrm{Hill}}}\xspace/H_{P,\,\textrm{PPD}}\xspace=2.4$; see Equation (\ref{['Gl:RHduerHPPPPPPa']})), either truncating the inflow approximately at the thickness of the CPD (black line) or allowing it down to the midplane (grey dashed). Only the gas between the pole ($\theta_0=0$) and $\theta_{0\rightarrow\textrm{p}}\xspace=8.3{\hbox{\textdegree}}\xspace$ (Equation (\ref{['Gl:mu0min']}), for the reference values in Equation (\ref{['Gl:RHduerHPPPPPPa']}) and $R_{\textrm{p}}\xspace=2~R_{\textrm{J}}\xspace$) hits the planet surface directly, with the rest landing on the CPD.
• Figure 3: Comparison of mass inflow rates ("accretion rates") into the Hill sphere from different work, normalised to the local value of $\Sigma_0\Omega{R_{\textrm{Hill}}}\xspace^2$: the actual rate in NGPPS (blue circles: $h\approx0.05$; grey: $h \BeginAccSupp{method=hex,unicode,ActualText=2249} \not\approx \EndAccSupp{}0.05$), the scaling from the ballistic model in this work (red diamonds), and the choksi23 "2D-Hill" scaling (yellow squares). The latter two need only the $q$ and $h$ values from the NGPPS planets as input into Equation (\ref{['Gl:MPktUebersicht']}) to determine the reduction of $\Sigma$. The boden13 scaling, derived for $h=0.05$, includes the effects of gap opening and is shown as black lines (solid for $\alpha=0.002$ and dashed for 0.004 and 0.01).
• Figure 4: Top: Planet growth rates in emsen21a for our selected planets at 1--5 Myr (black to red). Dashed lines show characteristic growth times of 0.1, 1, 10 Myr. Bottom: Gas surface density at the location of each planet, chosen based on the $q$ and $h$ parameters of the emsen21a so that our $\dot{M}_{\textrm{Hill,\,net}}$ equal the planet growth rate in the population synthesis $\dot{M}_{\textrm{pop synth}}$ (Equation (\ref{['Gl:SigSkal']})). Colours indicate $\log\dot{M}_{\textrm{pop synth}}\xspace/(M_{\textrm{J}}\xspace\,\textrm{yr}^{-1}\xspace)$ in bins with edges at $\{\infty,-5,-5.5,\ldots,-7.5,-\infty\}$ (dark red, salmon, lilac, purple, dusty blue, grey, pale grey). Short line segments on the right indicate the $\Sigma$ for which the vertical extinction $A=1$ or 0.3 mag when $\kappa=10~\textrm{cm}^2\,\textrm{g}^{-1}_{\textrm{gas}}\xspace$ (see text).
• Figure 5: Velocity of the preshock gas (left axes) and cumulative H $\alpha$ luminosity (right axes) for the shock at the surface at the planet (left panel) and of the CPD (right) for 2 $M_{\textrm{J}}$ (5 $M_{\textrm{J}}$) top (bottom) row. We set $\Sigma\approx0.02~\textrm{g}\,\textrm{cm}^{-2}\xspace$ and ${f_{\textrm{cent}}}\xspace\approx0.03$ to match the $\dot{M}_{\textrm{Hill,\,net}}$ and $\dot{M}_{\textrm{p,\,direct}}$ of the detailed simulations (see Section \ref{['Th:fzent']}). With this, the ballistic model of this paper (coloured lines) reproduces excellently the radiation-hydrodynamical simulations of m22Schock (grey points).