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Planetary Wind and Mass Loss Rate for HD209458b

1. AstroBEAR code and Set-up

In this study we use the AstroBEAR code (Cunningham et al 2009) to perform 3D hydrodynamic and magnetohydrodynamic numerical simulations and model the "Hot Jupiter" HD209458b ( Ballister et al 2007). AstroBEAR is a fully parallelized AMR MHD multi-physics code which currently includes modules for the treatment of self-gravity, ionization dynamics, chemistry, heat conduction, viscosity, resistivity and radiation transport via flux-limited diffusion. For our simulations we use a polytropic equations of state (the polytropic index \gamma is an input parameter) and we assume isothermal conditions \gamma=1.01

In this part we only focus on the planetary wind (hydrodynamic) for HD209458b without considering the star and stellar wind. We present the simulation results of planetary wind launching using the AstroBEAR code and calculate the mass loss rate of the planet using the density and velocity from the simulation data.

2. Parameters and Initial Conditions

The mass for H209458b is 0.69 M_{J} where M_{J} is the Jupiter mass (Wang et al 2002) and the radius is 1.38 R_{J} where R_{J} is the Jupiter radius (SouthWorth et al 2010). We use T_{p}=10^{4}K for the temperature of the planet.

\lambda=\frac{GM_{p}\mu}{R_{p}\kappa T_{p}}=10.74

\lambda measures the strength of the planetary wind. For this \lambda, a Parker-type thermally driven hydrodynamic wind is expected.As a comparason, the sun with its T\approx 10^{6} K corona has \lambda\approx 15.

We use 10^{-15} g/cm^{3} as the initial density for the planet atmosphere. For the initial temperature, we use two set-ups: 1) set the outer boundary of the planet with temperature T_{p}=10^{4}K (without temperature profile or spherically-launching wind) and 2) set the outer boundary of the planet with azimuthally variable temperature T(\theta)=T_{p}sin(\theta) where \theta=90^{\circ} is the sub solar point and T_{0}=T_{p}=10^{4}K (with temperature profile). For the 2nd case, we use similar initial set up to that of Stone & Progra (2009). We summarize the parameters we use for HD209458b are shown in Table 1.

M_{p}	0.69 M_{J}
R_{p}	1.38 R_{J}
T_{p}	10^{4} K
\rho_{p}	10^{-15} g/cm^{3}
\lambda_{p}	10.74

Table 1. Parameters for HD209458b

3. Resolutions

In our simulations the planet is considered as an internal boundary and the physical quantities are fixed during the simulation. Our computational domain consists a cube of size \approx 29R_{p} with resolution 40^3 for the base grid and totally 6-level of AMR is used. This makes the finest resolution up to \approx 88 zones per R_{p}.

4. Planetary Wind Results and Mass Loss Rate

In Figure 1, we show the 3D simulation results for both without-temperature-profile and with-temperature-profile cases. For the without-temperature-profile case (top panels in Fig.1), we can see the planet temperature launches a spherical thermal wind and Mach=1 contour is approximately spherical or circle in 2D cross section. While for the with-temperature-profile case (bottom panels in Fig. 1), there's flow across from the dayside to the nightside and the Mach=1 contour shows there's a weak shock between two sides.

in color
in gray

Fig. 1 Steady state planetary wind solution of cross section in the xy-plane for simulations without (top) and with the temperature profile. Flow and density are shown on the left and thermal structure and M=1 contours are shown on the right. The small circle at the center shows the radius of the planet.

The mass loss rate can be calculated by integrating \oint \rho\mathbf{v}\cdot d\mathbf{s} From our 3D simulation data, the mass loss rate for the without-the-temperature-profile case is 2.20\times10^{9} g/s comparing with 1.57\times 10^{9} g/s for the with-the-temperature-profile case. We see the sin(\theta) temperature profile makes a mass loss about 70\% of the value for the spherically-launching wind.

With the planet temperature T_{p}=10^{4}K, we can also analytically solve the problem in 1D with Parker's wind solution (Parker 1958). The mass loss rate with Parker's wind solution gives 3.11\times10^{9} g/s. The estimated mass loss rate for H209458b can be found in Table 2.

Methods	Mass Loss Rate
3D Simulation Without Temperature Profile	2.20\times 10^{9} g/s
3D Simulation With Temperature Profile	1.57\times 10^{9} g/s
Analytic Parker wind Solution	3.11\times 10^{9} g/s

Table 2. Estimated Mass Loss Rate for HD209458b

5. References

Ballister, G., King, D., & Herbert, 2007, Nature, 445, 511

Cunningham A., Frank, A., Varniere, P., Mitran, S., Jones, T. W., 2009, ApJS, 182, 519

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