tokens: on different methods. A simple radial profile (black line within gray boundaries) nicely fits our results and the previous estimates in the range 1.1−2.0 R ⊙ . 3. Coronal magnetic field The upstream Alfvénic Mach number M A and the magnetic field strength B can be derived from the standoff distance Δ r ≡ r sh − r cme between the shock front r sh and the CME leading edge r cme by employing a model adapted by Gopalswamy Yashiro (2011) but originally developed for the bow shock of the Earth. Physically, the shock Mach number M and the radius of curvature r cme of the nose of the driver control the standoff distance. The ratio δ between Δ r and r cme can be expressed as δ ≡ Δ r / r cme = 0.81[( γ − 1) M 2 + 2]/[( γ − 1)( M 2 − 1)], (e.g., Farris Russell 1994 ) where γ is the adiabatic index (we assume γ = 4/3 and γ = 5/3 as two possible values). This expression allows the applicability of the model to the low- M regime provided that M is substituted for the fast magnetosonic Mach number ( Fairfield et al. 2001 ). In the inner corona, v A substantially exceeds the sound speed meaning that M can be confidently replaced by M A . It then follows that , implying that M A (and thus the coronal magnetic field strength B , given the upstream electron density profile n e ) can be obtained at any given time by measuring δ . We point out that the above method is based on a model by Farris Russell (1994) , further extended by Russell Mulligan (2002) , which was specifically designed for the bow shock of the terrestrial magnetosphere. As such, the evaluation of the standoff distance in that model is intended to be strictly valid for the nose of a CME. It is then important to discuss how appropriate such a model is for a coronal CME-driven shock, for which, moreover, as in our case, the standoff distance is calculated not at the nose but at the flanks of the shock. In order to check the validity of the model in the coronal environment, Schmidt et al. (2016) used a 3D magnetohydrodynamic code to test the accuracy of the Gopalswamy Yashiro (2011) method by applying it to a simulation of a real CME event, where the standoff distances between the driven shock and the CME could be determined within the simulation box. As a result, the predicted magnetic field strength profile agreed very well (within 30% between 1.8 and 10 R ⊙ ) with the magnetic field strength profile found in the simulation domain, thus providing strong support for the Gopalswamy Yashiro (2011) method in a coronal environment. As for the second issue, that is, the reliability of the model when the standoff distance is evaluated at the flanks of the shock surface, it is obvious that in the general case of a fast moving CME the shock surface would be better represented by a 3D oblate surface. In that case, our model would yield unreliable results away from the CME nose. In our specific case however, since the translational motion of the CME away from the Sun is negligible with respect to its
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