### Current Research

**Overshooting in the Upper Main Sequence**

Collaborators: Casey Meakin(LANL and UA), Justin Brown (UCSC), Stan Woosley(UCSC)

Our distant goal in this project is to tackle the problem of mixing at the boundaries in 1D calculations. Traditionally such situations are handled with the "step" or exponential overshooting prescriptions within the Mixing Length Theory (MLT) formalism. Since these phenomenological appoaches are not really based on physics, the associated free-parameters are not observationally well constrained. For instance, the tuning parameter of "step" overshooting looks like this:

Figure 1 of Meakin et al. (2011) - overshoot parameter estimates from dEBs (diamonds) and asteroseismological data (triangles). The values adopted by Girardi et al. (2000) and Pietrinferni et al. (2004) for populations synthesis are labeled G2000 and P2004. The mass uncertainties for the eclipsing binary data is negligible on the log(M) scale used. Overshoot parameters inferred from asteroseismic data are from Soriano & Vauclair (2010), Di Mauro et al. (2003), Sua´rez et al. (2009), Briquet et al. (2007), Dupret et al. (2004), Aerts et al. (2006), Desmet et al. (2009), and Mazumdar et al. (2006); and the values from eclipsing binary data are from Claret (2007).

In this study we are particularly looking at the convective core overshooting problem. We are implementing various physics based prescriptions (from Rempel (2004), Meakin et al.(2007)) in both 1D hydrodynamical code KEPLER and open source stellar evolutionary code MESA, and aiming to perform a detailed comparison with high quality data from dEBs (e.g. Torres et al. (2010)). A rough summary of this work is discussed in this paper, and a detailed paper is in preparation.

**Semi-Analytic Description of Carbon Burning Episodes**

Collaborators: Stan Woosley(UCSC), Mark Krumholz(UCSC).

It is a well known fact that for solar metallicity massive stars at around 20Msun carbon starts to burn radiatively in the core rather than convectively. A beautiful semi-analytic desctiption of this is given in Barkat (1994). Though many studies mention this feature, almost all of them completely ignore implications from the intricate interplay between the resulting carbon burning shells and oxygen burning core. As found in Sukhbold and Woosley (2013), this interplay is the main reason behind the non-monotonic presupernova core structure.

Convection plot of a 19.9Msun solar metallicity KEPLER model from the S-series data of Sukhbold and Woosley (2013). Notice the 1st carbon burning shell has become mostly radiative.

In this work we are am aiming to develop a complete semi-analytic description of the major carbon burning episodes in 1D calculations.

**Compactness of Presupernova Stellar Cores**

Collaborators: Stan Woosley(UCSC), Alex Heger (Monash), Bill Paxton (UCSB).

The compactness of the core of a massive star is basically a measure of the density gradient in the iron core and the silicon and oxygen shells surrounding it. It varies non-monotonically as a function of initial mass, and the connection between this parameter and explodability due to standard neutrino-transport mechanism has been noted several times recently (see O'Connor and Ott 2011, 2013; Ugliano et al. (2012)).

Figure 2 of Sukhbold and Woosley (2013). The compactness parameter as defined in the O'Connor and Ott (2011), calculated for 151 KEPLER models in the mass range of 15-30 Msol

However, none of the previous studies explain why it is non-monotonic, how robust the results are and its sensitivity to uncertain input physics. In this work we exactly address these issues in terms of stellar physics using a large set of full star and bare CO core calculations with KEPLER and MESA. The results of this work is submitted to ApJ.

**Analytic Solution to the Lane-Emden Equation**

Collaborators: Hassen Yesuf (UCSC)

As we all know form undergraduate physics class, the Lane-Emden equation basically describes the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid. Its a special case of the elliptic type PDE, connecting the potential with the density, otherwise known as Poisson's equation. Polytropes of n=0,1,5 can be solved analytically (Chandrasekhar 1967, p. 91):

In this work, with my friend Hassen, we're trying to find analytic solutions for the polytropic indexes of n=2,3,4.