LYα TRANSIT SPECTROSCOPY AND THE NEUTRAL HYDROGEN TAIL OF THE HOT NEPTUNE GJ 436b

A promising exoplanet for UV transit observations is GJ 436b, a hot Neptune, (M = 0.07 M_Jup), orbiting an M2 dwarf at 0.029 AU with a period of 2.6 days. GJ 436b is particularly promising because it is very close to Earth, at a distance of only 10.2 pc. The properties of the system are summarized in Table 1. GJ 436b was discovered by Butler et al. (2004) using the radial velocity method, but was later found to be transiting its host star (Gillon et al. 2007).

Table 1. Properties of GJ 436 and GJ 436b

Property	Value	Reference
Host star spectral type	M2 V	Butler et al. (2004)
Distance (pc)	$10.14^{+0.25}_{-0.23}$	van Leeuwen (2007)
M*/M☉	$0.452^{+0.014}_{-0.012}$	Torres et al. (2008)
R*/R☉	$0.464^{+0.009}_{-0.011}$	Torres et al. (2008)
Porbit(days)	2.643850 ± 9 × 10−5	Pont et al. (2009)
Transit center (JD)	2454279.436714 ± 1.5 × 10−5	Pont et al. (2009)
R.A. (h:m:s)	+11:42:11.18	Zacharias et al. (2012)
Decl. (d:m:s)	+26:42:22.64	Zacharias et al. (2012)
Rplanet/RJup	$0.3767^{+0.0082}_{-0.0092}$	Torres et al. (2008)
Mplanet/MJup	0.0727 ± 0.0032	Butler et al. (2006)
Semimajor axis (AU)	0.02872 ± 0.0048	Butler et al. (2006)
Transit duration (hr)	0.7608 ± 0.012	Pont et al. (2009)
Transit depth for Rplanet	0.00696 ± 0.000117	Torres et al. (2008)
Escape speed from GJ 436b (km s−1)	26.4
Orbital velocity amplitude (km s−1)	118
Radial velocity (km s−1)	9.6 ± 0.1	Nidever et al. (2002)

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In 2010 Spitzer observed GJ 436 during several secondary transits in six bands from 3.6 to 24 μm. Using a Metropolis–Hastings Markov-chain Monte Carlo (MCMC) model, Stevenson et al. (2010) explored a wide range of parameter space to determine the best-fit compositional models. They found a high CO abundance and a deficiency of CH₄ relative to thermochemical equilibrium. Madhusudhan & Seager (2011) also used MCMC to confirm the overabundance of CO and CO₂, and a slight underabundance of H₂O, as compared to equilibrium chemistry with solar metallicity. They explained the observed abundances by a combination of high metallicity (∼10 × solar) and vertical mixing. Observations by Knutson et al. (2014) indicate an effectively featureless transmission spectrum, ruling out cloud-free, hydrogen-dominated atmosphere models. The measured spectrum is consistent with either a high cloud or haze layer or with a relatively hydrogen-poor atmospheric composition. Hu & Seager (2014) find that hot Neptunes, like GJ 436b, are likely to have thick atmospheres that are not hydrogen dominated, but are water-rich or hydrocarbon-rich depending on their C/O ratio. Using limited Hubble Space Telescope (HST)/Space Telescope Imaging Spectrograph (STIS) data of the stellar Lyα flux, Ehrenreich et al. (2011) developed numerical simulations to determine the transit signature of GJ 436b for various assumed mass-loss rates. They predicted an 11% transit depth in Lyα for a mass-loss rate of 10¹⁰ g s⁻¹. The analysis we have conducted is the first to place observational constraints on the mass-loss rate of GJ 436b or any other Neptune-mass exoplanet.

In this paper, we present and analyze transit observations of GJ 436b in Lyα observed with STIS on the HST. In Section 2 we describe the data sets used and the reduction process, and how we created the lightcurves and velocity profiles. We present the lightcurve and velocity profile for Lyα in Section 3 along with the measured absorption depths. In Section 4 we discuss the structure of the system and calculate a range of mass-loss rates for GJ 436b. We summarize our results in Section 5.

The HST/STIS transit data of GJ 436 were obtained with the G140M grating using a long slit with dimensions 52'' × 01. We selected a central wavelength of 1222 Å, covering the spectral range of 1194–1249 Å with a spectral resolution of about 25 km s⁻¹. The data were taken with the FUV-MAMA detector using the time-tag mode. Table 2 summarizes all of the observations. Our four orbit per transit observing cadence is comparable to that used by Lecavelier des Etangs et al. (2012) for velocity resolved observations of the extended hydrogen atmosphere of the hot Jupiter HD 189733b with the same STIS grating configuration. Given an error of 2%, the relative photometric accuracy of STIS, issues of persistence and instrumental settling are less significant when observing the 10% UV transit signal of the exosphere compared to the <1% near-IR molecular diagnostics of the lower atmosphere.

We reduced the GJ 436 data using CALSTIS v2.40 (2012 May 23). For the final two exposures, the CALSTIS pipeline did not correctly extract the one-dimensional spectrum (the "x1d" file) from the two-dimensional data. We therefore manually extracted the one-dimensional spectrum for all four exposures. Since in the "x2d" file, the source was located at pixels 481–491, we subtracted a background ribbon at pixels 492–502 from the source data (see Figure 1). A wavelength solution was then calculated by using the reference wavelength, reference pixel, and dispersion from the header. Finally, we obtained fluxes by multiplying by the angular area covered by the pixels and an additional scale factor (determined empirically by matching to a correctly extracted spectrum from the "x1d" files for the first two exposures).

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To characterize the transit, we take two complementary approaches. The first is the creation and analysis of lightcurves. This approach allows us to study the transit with higher time resolution. The lightcurve creation method is described in detail below. In the second method, we analyze the spectral difference between the pre-ingress, transit, and post-egress data. This method maintains the velocity resolution of the data. We also created velocity profiles for Lyα. We compared the pre-ingress spectrum to the transit and post-egress spectra as a function of velocity. We also looked at the difference between the spectra (pre-ingress minus post-egress, such that absorption is a positive difference) versus velocity. We then calculated the post-egress depth by integrating this difference spectrum.

2.2.1. Light Curve Creation

We show the Lyα lightcurve for GJ 436 in Figure 2 and the Lα velocity profile in Figure 3. Table 4 shows the transit depths extracted from these figures. The transit depths are identical for the two procedures (because the reference spectrum and the integration method are the same), however, the spectral analysis results in somewhat larger errors, due to the extra step of subtracting prior to normalizing and integrating. We will use the higher time resolution lightcurve data to analyze the transit depth. For GJ 436b we see a mid-transit depth of 16.6% ± 7.2% in the blue wing and 4.5% ± 5.7% in the red wing. This corresponds to an occulting disk of 5.0 R_p and 2.6 R_p respectively, both smaller than the Roche lobe radius of 6.1 R_p. When both wings are combined, the mid-transit depth is 8.8% ± 4.5%, corresponding to an equivalent opaque occulting disk of 3.6 R_p. The asymmetry in the absorption can be explained by charge exchange of the stellar wind with the atmosphere. As viewed from Earth, only the stellar wind traveling toward us can be observed, causing excess absorption blue-ward of line center. This type of asymmetry, with more absorption in the blue wing, is also seen in the Lyα absorption during transit from HD 209458b (Vidal-Madjar et al. 2003) and HD 189733b (Lecavelier des Etangs et al. 2012).

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Interestingly, the Lyα transit extends much later in phase than the optical transit. We examine these data for the possibility of extended egress, finding post-egress depths of 29.9%  ±  6.4% in the blue wing, 19.1%  ±  5.0% in the red wing, and 22.9%  ±  3.9% for both wings combined ∼2 hr after mid-transit and about 1.5 hr after fourth contact. The time-tag data points are presented in Table 5. These data corroborate the detection of both the transit and extended egress, as three of the four transit data points and all of the post-egress data points show a transit detection, although large variations in the blue-wing time-tag data are observed near mid-transit.

To determine whether or not the extended egress is real, we consider the time-tag data. We form the null hypothesis that the data are Gaussian distributed and the apparent occultation is random noise. We use the average of the error bars on the time-tag data as the standard deviation for the Gaussian distributions (σ_blue = 0.1092, σ_red = 0.08930), both with a mean of unity. Assuming these distributions, we determined the probability of finding four consecutive points lower than the highest point in the set of four at the deepest transit depth, which is located at phase ∼2 hr (x_blue = 0.3003, x_red = 0.1187 below the mean). We did this by randomly picking 14 points from the specified Gaussian distribution and counting how many trials out of 10⁷ had four consecutive points outside the requisite range. For the parameters determined for the blue wing, none of the 10⁷ trials had four consecutive points. For the red wing distribution, we found a probability of 9.31 ± 0.76 × 10⁻⁵ to randomly produce the result. We therefore conclude that the deep, extended egress signal seen in Figure 2 is real and not due to random statistical variations.

While we have determined that the post-egress detection is not due to statistical noise, we have not yet considered whether the drop in flux could be due to stellar variability, as opposed to atmospheric absorption from GJ 436b. To address this issue, we look at a resonance line from N v, an ion that we do not expect to find in the atmosphere of the exoplanet. For the N v time-tag data points we find rms=0.2765, while the average value of the 1σ error bar for those points is 0.4010. We conclude that the signal to noise is too low for the N v doublet to be a suitable tracer of stellar variability. Similarly, there was not enough flux in the Si iii line to be measurable above the noise.

Instead we look to the literature to assess the potential magnitude of stellar variability. Loyd & France (2014) studied time variability in the C ii, Si iii, and Si iv resonance lines of 38 cool stars, including GJ 436. They did not attempt to characterize Lyα line variability because geocoronal airglow cannot be removed reliably from their Cosmic Origins Spectrograph data. Instead we use their C ii emission line variability, because C ii has a similar formation temperature to Lyα (T_form ≈ (1–3) × 10⁴ K; Dere et al. 2009). Loyd & France (2014) found that the mean-normalized chromospheric C ii line variability, excluding flare periods, in GJ 436 is $0.20^{0.09}_{0.12}$ on 60 s timescales. Our post-egress data cover a 48 minute time span. Assuming that the stellar variability is uncorrelated over time, the noise associated with stellar fluctuations is estimated to be $0.20/\sqrt{48}=0.0289$. Combining the eight post-egress time-tag data points and the photon noise with the upper limit on the noise expected from chromospheric variability, we find a 23.7% occultation with an uncertainty of 4.5%. From this we conclude that the post-egress detection is very likely real. Future observations over several transit cycles would be very valuable.

Some models of the interaction between escaping gas from the exoplanet's atmosphere and the stellar wind predict a comet-like tail extending behind the planet (Schneider et al. 1998; Bourrier & Lecavelier des Etangs 2013). The Lyα data support this type of structure in the GJ 436 system. A large, inflated cloud of gas trailing the planet could explain an occultation deeper than that observed in the optical and post-egress absorption long after optical transit.

In order to simulate the structure of the exoplanetary gas cloud at the orbit of GJ 436b, we attenuate the pre-ingress profile by a column of hydrogen and deuterium to match the post-egress spectrum. We search for a χ² best fit neutral hydrogen column ([log(N_H i)] ranging from 14.0 to 19.0 in step sizes of 0.2 dex) for a grid of stellar covering fraction (assuming uniform optical thickness of the gas over the area covered and uniform emission of Lyα from the stellar disk) versus Doppler b value, $b=(({2kT}/{m_{{\rm H}}})+v_{{\rm turb}}^2)^{1/2}$. We assume a fixed D/H ratio of 1.5 × 10⁻⁵ (Linsky et al. 2006). For the range of tested covering fractions and b values, we find two parameter regimes that provide a reasonable fit to the data. The first is a low covering fraction, high N_H i model. The second is a high covering fraction low N_H i model. We expect the second regime to be more physically plausible. The low covering fraction models require N_H i ∼ 10¹⁹ cm⁻². Assuming that the comet-like tail extends to a few times the orbital radius, ∼0.1 AU in the line of sight (LOS) to the star, we find a number density n_H  ∼  few × 10⁷ cm⁻³. This density is far above the estimates (∼3 × 10⁵ cm⁻³) from comet-like tail models (Bourrier & Lecavelier des Etangs 2013). A column density of N_H i = 10¹⁵ cm⁻², typical of the high covering fraction regime, converts to a number density n_H i ∼ few × 10⁴ cm⁻³.

Taking the high covering fraction and low N_H i case as more plausible, acceptable fits to the post-egress Lyα spectrum require the Doppler b-parameter to be between 60 and 120 km s⁻¹. We require b values in this range to fit the shape of the post-egress Lyα profile. For a typical atmospheric temperature of 10⁴ K, the thermal width is 13 km s⁻¹, requiring a superthermal velocity component of ∼50–110 km s⁻¹ to explain such high b values. Charge exchange between stellar wind protons and planetary wind neutral H atoms will lead to neutral H atoms with high velocities. For a representative covering fraction of 0.8, we can then limit log(N_H i) to 14–16. Figure 4 shows an example best fit attenuation profile with parameters within this range. These ranges are shown in Table 6.

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To achieve a covering fraction of 0.8 requires a cloud of R ≈ 10R_planet. Using the column density and b (to calculate the cross-section at the line core, $\sigma =({\sqrt{\pi }e^2}/{m_ec})({f\lambda _0}/{b})$), we can estimate the line center optical depth of the extended neutral hydrogen atmosphere (τ₀ = N_Hσ). For the ranges of N_H and b, we find τ₀ = 0.63–130. Koskinen et al. (2010) predict the hydrogen cloud to be optically thick in the line wings, indicating that our models with the lowest columns and higher b (which give the higher τ_νs) are more physically representative.

4.2.1. Spherical Mass-loss

We first calculate the mass-loss rate for GJ 436b by assuming a spherically symmetric envelope around the planet that blocks a portion of the stellar surface. We consider an LOS toward the center of the star that passes through the exoplanet's atmosphere a distance p from the center of the planet. We measure x along the LOS and r along from the center of the planet (see Figure 5). For a spherical outflow with a constant mass-loss rate, the mass flux in neutral hydrogen from the planet is

$$\begin{equation} \dot{M}_{{\rm H\,\scriptsize{I}}} =4\pi r^2 m_{{\rm H}} v n_{{\rm H\,\scriptsize{I}}}(r), \end{equation} \tag{ 1 }$$

where v is the outflow velocity at r. The optical depth at the line center along this LOS is

$$\begin{equation} \tau _0 =\sigma \int _0^{\infty } n_{{\rm H\,\scriptsize{I}}}(x)dx, \end{equation} \tag{ 2 }$$

where n_H i is the number density of neutral hydrogen, σ is the absorption cross section. Combining Equations (1) and (2) yields

$$\begin{equation} \tau _0=\frac{\sigma \dot{M}_{{\rm H\,\scriptsize{I}}}}{4\pi m_{{\rm H}}}\int _0^\infty \frac{dx}{vr^2} \Rightarrow \dot{M}_{{\rm H\,\scriptsize{I}}}=\frac{4\pi m_{{\rm H}} \tau _0}{\sigma } \left[ \int _0^\infty \frac{dx}{vr^2} \right]^{-1}. \end{equation} \tag{ 3 }$$

In the integrand, the r⁻² factor gives the highest weight to v values where the LOS passes closest to the planet (smallest r). Thus, for an order-of-magnitude calculation v may be taken out of the integral and replaced with a value representative of that expected near the radius at closest approach, r = p, between the LOS and planet. (This assumes v does not drop precipitously at large r, consistent with the models of Murray-Clay et al. 2009.) Bringing v out of the integral, we are able to find an analytical solution. From Figure 5 we can use r² = (x − a)² + p², where a is the star–planet distance, to evaluate the integral in Equation (3).

$$\begin{equation} \int _0^\infty \frac{dx}{r^2} = \int _0^\infty \frac{dx}{(x-a)^2+p^2} = \frac{\pi }{2p}+\frac{1}{p}\arctan \left(\frac{a}{p}\right). \end{equation} \tag{ 4 }$$

Since only LOSs that intersect the stellar disk, p < R_⋆, are of interest, and for GJ 436 a/R_⋆ = 13.34, $\arctan (a/p)$ may be taken to be π/2 to good accuracy, so that the integral in Equation (4) simplifies to π/p. Thus

$$\begin{equation} \dot{M}_{{\rm H\,\scriptsize{I}}}\approx \frac{4pvm_{{\rm H}}\tau _0}{\sigma _0}. \end{equation} \tag{ 5 }$$

The cross section at line center is given by

$$\begin{equation} \sigma _0=\frac{\sqrt{\pi }e^2}{m_ec\Delta \nu _D}f \end{equation} \tag{ 6 }$$

f = 0.4161 is the oscillator strength, m_e is the electron mass, and Δν_D is the Doppler width given by

$$\begin{equation*} \nonumber \Delta \nu _D=\frac{\nu _0}{c} \left(\frac{2kT}{m_{{\rm H}}}+v_{{\rm turb}}^2 \right)^{1/2}=\frac{b}{\lambda _0}. \end{equation*}$$

Given the range of b values determined in Section 4.1 (b = 60–120 km s⁻¹), we limit the cross section at Lyα to σ₀(H i) = 6.3 × 10⁻¹⁵–1.3 × 10⁻¹⁴.

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Assuming the absorption is due to an optically thin cloud covering the star, we consider annuli of area 2πpdp centered on the planet. Each annulus absorbs a fraction of the light from the stellar disk equal to

$$\begin{equation} \frac{dF_\star }{F_\star }=\frac{2\pi p dp (1-e^{-\tau _0})}{\pi R_\star ^2}=\frac{2}{R_\star ^2}p(1-e^{-\tau _0})dp, \end{equation} \tag{ 7 }$$

where τ₀ = τ₀(p) is the optical depth at line center along the LOS passing through the annulus of radius p. The observed transit depth (δ) is then

$$\begin{equation} \delta =\int _{R_p}^{R_\star }\frac{dF}{F}=\frac{2}{R_\star ^2}\int _{R_p}^{R_\star }p(1-e^{-\tau _0})dp. \end{equation} \tag{ 8 }$$

According to Murray-Clay et al. (2009), τ₀ < 0.1 for p ≳ 1.5R_p, so using the small τ approximation, Equation (8) becomes

$$\begin{equation} \delta =\frac{2}{R_\star ^2}\int _{R_p}^{R_\star }p\tau _0dp. \end{equation} \tag{ 9 }$$

Using Equation (5) this becomes

$$\begin{equation} \delta \approx \frac{2}{R_\star ^2}\frac{\dot{M}\sigma _0}{4m}\int _{R_p}^{R_\star }\frac{dp}{v} \approx \frac{2}{R_\star ^2}\frac{\dot{M}\sigma _0}{4mv}(R_\star -R_p) \approx \frac{2}{R_\star }\frac{\dot{M}\sigma _0}{4mv}. \end{equation} \tag{ 10 }$$

While Murray-Clay et al. (2009) predict v to vary by a factor of a few over the range of p considered here, we have approximated v to be constant with p. We have also neglected the R_p ≪ R_⋆ term. Solving for the mass loss rate, we get

$$\begin{equation} \dot{M}\approx \frac{2\delta R_\star m v}{\sigma _0}. \end{equation} \tag{ 11 }$$

Following Murray-Clay et al. (2009), we adopt v = 10 km s⁻¹, approximately equal to the thermal velocity of hydrogen at 10⁴ K. We use the observed mid-transit depth δ = 0.088  ±  0.045. Thus, these observations, for our range of b values, bound the mass-loss rate in neutral hydrogen to $\dot{M}_{{\rm H\,\scriptsize{I}}}=3.7 \times 10^5\hbox{--}2.3 \times 10^6 \,\mathrm{g\,s}^{-1}$.

The mass-loss rate depends linearly on the b value. So, if the b value at the exobase, where the wind is launched, differs from b values we have calculated, the mass-loss rate would differ correspondingly. Because the planet is so close to its host star, the EUV flux and charge exchange with the stellar wind will ionize most of the escaping hydrogen and only a small fraction will be neutral in the outer atmosphere. Koskinen et al. (2013) model the H and H⁺ density in the atmosphere of HD 209458b as a function of planetary radius. In the upper atmosphere (R ∼ 5R_p) the neutral fraction is ∼0.1. Assuming similar ionization conditions in the extended atmosphere of GJ 436b, we correct for this neutral fraction, and calculate a total mass-loss rate

$$\begin{equation} \dot{M}=3.7 \times 10^6\hbox{--}2.3 \times 10^{7} \,\mathrm{g\,s}^{-1}. \end{equation} \tag{ 12 }$$

Assuming the structure of GJ 436b is similar to that of Neptune, and the atmosphere comprises 5%–15% of the mass of the planet (Guillot 1999), and that the mass-loss rate is constant in time, this range of mass-loss rates give a range of atmospheric lifetimes of 9.5 × 10¹²–1.8 × 10¹⁴ yr, indicating that the atmosphere is stable over the lifetime of the star.

4.2.2. EUV Heating to PdV Work

We also calculate the mass-loss rate following the analytical argument of Murray-Clay et al. (2009). We first calculate the amount of stellar EUV flux available to heat the atmosphere,

$$\begin{equation} E_{{\rm heat}}=\epsilon \pi F_{{\rm EUV}} R_p^2 \;\, (\mathrm{erg\,s}^{-1}). \end{equation} \tag{ 13 }$$

Here is an efficiency factor, F_EUV (in erg cm⁻² s⁻¹) is the stellar flux at the orbit of the planet from 300 to 912 Å (where the photoionization cross-section for hydrogen is largest; Murray-Clay et al. 2009), and R_p is the planetary radius (R_p = 0.38R_Jup). Using the model scaling relations of Linsky et al. (2014), we estimated the EUV flux based on the reconstructed Lyα luminosity (France et al. 2013). These calculated EUV luminosities are shown in Table 7. We then consider the PdV work required to liberate a unit mass from the gravitational well of the planet:

$$\begin{equation} \frac{P\Delta V}{\rho R_p^2 H}\sim \frac{P R_p^3}{\rho R_p^2 H}\sim \frac{\rho g H R_p^3}{\rho R_p^2 H}\sim \frac{{G M}_p}{R_p}. \end{equation} \tag{ 14 }$$

The mass-loss rate is then the ratio of the heat available to the work required to lift out mass:

$$\begin{equation} \dot{M}=\frac{\epsilon \pi F_{{\rm EUV}} R_p^2}{{ G M}_p/R_p} = \frac{\epsilon \pi F_{{\rm EUV}}R_p^3}{{ G M}_p} =1.1\times 10^9 \,\mathrm{g\,s^{-1},} \end{equation} \tag{ 15 }$$

corresponding to a lifetime of 4 × 10¹¹ yr. Here we have assumed an efficiency = 0.3 and extrapolated an EUV flux F_EUV = 607 erg cm⁻² s⁻¹ from the reconstructed intrinsic Lyα flux. This argument is valid only if we are in the EUV driven, as opposed to X-ray driven, evaporation regime. According to Owen & Jackson (2012) for a Neptune-mass planet with a density of 1 g cm⁻³ at a separation of 0.025 AU (for GJ 436b: mass = 1.35 M_Nep, density = 1.69 g cm⁻³, separation = 0.029 AU) the critical X-ray luminosity at which the planetary wind will transition from X-ray driven to EUV driven is ∼8 × 10²⁸ erg s⁻¹. Kashyap et al. (2008) measure the X-ray luminosity of GJ 436 to be 1.4 × 10²⁷ erg s⁻¹, two orders of magnitude lower than the transition value, placing GJ 436b well into the EUV driven evaporation regime.

Table 7. EUV Luminosity of GJ 436

Band	Luminosity
	(erg s−1)
<100 Åa	1.45 × 1027
100–200 Å	1.44 × 1027
200–300 Å	1.26 × 1027
300–400 Å	1.12 × 1027
400–500 Å	2.48 × 1025
500–600 Å	4.47 × 1025
600–700 Å	5.82 × 1025
700–800 Å	6.98 × 1025
800–912 Å	9.41 × 1025

Note. ^a5–124 Å luminosity from Kashyap et al. (2008).

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These two methods give results that differ by two orders of magnitude. This is not surprising given that mass-loss calculations for HD 209458 can vary by four orders of magnitude depending on method. Our range of mass-loss values are roughly consistent with the models of Ehrenreich et al. (2011). Their model transit curves can be seen in Figure 2. They predict a transit depth of 11% for a mass-loss rate of 10¹⁰ g s⁻¹. Their calculation of the mass-loss rate was based on the measured stellar X-ray (5–124 Å) luminosity, $\log (L_{X} \,({\rm erg\,s}^{-1}))=26.85^{+0.65}_{-0.89}$ (Hünsch et al. 1999). However, this X-ray luminosity arises in the stellar corona, and may not be representative of the majority of the longer wavelength EUV flux from GJ 436, which is likely dominated by Lyman continuum emission from the transition region and upper chromosphere (Linsky et al. 2014). We find EUV luminosities lower than Ehrenreich et al. (2011), especially at wavelengths where the photoionization cross-section for hydrogen is largest (300–900 Å; Murray-Clay et al. 2009). Their calculation also depends linearly on the heating efficiency (η, similar to above) in the atmosphere, a parameter that is not well constrained. For η = 0.15 they calculate $\dot{M}=1.60\times 10^9$ g s⁻¹, but this mass-loss rate can range between 1.07 × 10⁸ g s⁻¹ and 1.07 × 10¹⁰ g s⁻¹ for η between 0.01 and 1. Uncertainties in the neutral fraction that we use in our first calculation, in our second calculation, and the parameters of the Ehrenreich et al. (2011) calculation leave a wide range for the possible mass-loss rate of GJ 436b.