3. Hot Spot Phase Shift on Tidally-locked Exoplanets. A full-globe map of the exoplanet HD 189733b reveals a 'hot spot t

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3 Hot Spot Phase Shift On Tidally Locked Exoplanets A Full Globe Map Of The Exoplanet Hd 189733b Reveals A Hot Spot T 1 (53.61 KiB) Viewed 69 times

3. Hot Spot Phase Shift on Tidally-locked Exoplanets. A full-globe map of the exoplanet HD 189733b reveals a 'hot spot that is offset from the substellar point by about 30 degrees (see the figure here, from H. Knutson). The offset may indicate eastward jet stream winds. Let us construct a simple radiative-dynamical model to understand this system. Assume that the longitudinal temperature distribution is governed by two processes: horizontal wind advection and radiative relaxationA full- globe map of the exoplanet HD 189733b reveals a 'hot spot that is offset from the substellar point by about 30 degrees (see the figure here, from H. Knutson). The offset may indicate eastward jet stream winds. Let us construct a simple radiative- dynamical model to understand this system. Assume that the longitudinal temperature distribution is governed by two processes: horizontal wind advection and radiative relaxation. In the longitude direction, the equation could be written as: at 1 JT + T(X)-T Trad at Tadd where T is temperature and A is longitude (2-0 is the substellar point and λ = It is the anti-stellar point). Te (2) is the temperature distribution under radiative equilibrium. We use a sinusoidal function that Teg (2)= To +47 cosa, where To is the global mean temperature of the atmosphere. 47 is the temperature difference between dayside and nightside. Tade = a/ū is the advection timescale where a is planetary radius and the mean zonal jet speed. Trad is the radiative relaxation timescale. Trad can be estimated using: Trad- PC₂ goT where P is the atmospheric pressure at the layer of hot spot phase shift, c, is the heat capacity, g is gravity, or is Stefan-Boltzmann constant. (a) Solve this system in steady state (what does the "steady state" mean when you solve the equation?). Obtain an analytical expression of temperature distribution as a function of longitude. (b) Show that the hot spot phase shift with respect to the substellar point is controlled by a single parameter e = Trad/Tade: Describe qualitatively intuitively how does the hot spot phase shift depend on the mean temperature To, day-night temperature difference AT and zonal jet speed Ü. (c) HD 189733b's radius is about 79,000 km, slightly larger than Jupiter. We assume the phase shift occurs at 50 mbar where T-1100 K. The heat capacity of hydrogen c₂ = 1.3 x 10 J kg ¹K¹, and gravity g is about 21 m s. If the hot spots shift is indeed 30 degree as observed, what is the eastward wind speed U on that planet? How does that compare with the wind speed on our Jupiter?