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For a brief summary of the capabilities of MESA, take a look at this presentation by Matteo Cantiello. (He has also provided a Keynote version.)
The full capabilities of MESA are documented in the instrument papers. Looking through the figures will give you a feel for the broad range of problems that can be studied using MESA.
Additional capabilities are documented by MESA users in their publications
To give you yet more flavor of what MESA can do, here are some movies made using png output from mesa/star.
This movie has more than you ever wanted to know about the evolution of a 1 Msun star from pre-main sequence to cool white dwarf. (61MB)
[![1 Msun from pre-main sequence to white dwarf][2]][1] [1]:assets/movies/1M_pre_ms_to_wd.mov [2]:assets/movies/1M_pre_ms_to_wd.png
Here is the evolution of a 2 Msun star from He core flash to cool white dwarf with some nice thermal pulses at the tip of the AGB (initial Z = 0.02). (40MB)
[![2 Msun from He flash to white dwarf][4]][3] [3]:assets/movies/2M_he_flash_to_wd.mov [4]:assets/movies/2M_he_flash_to_wd.png
This one shows the evolution of a 16 Msun star from zams to core collapse (initial Z = 0.02 and ZAMS uniform rotation = 0.5 of critical). (13MB)
[![16 Msun from ZAMS to core collapse][6]][5] [5]:assets/movies/16M_z2m2_wwc70.mov [6]:assets/movies/16M_z2m2_wwc70.png
Now for something completely different, here is a movie showing rapid accretion of pure helium onto a neutron star envelope leading to Type I X-ray bursts. (10MB)
[![Type I X-ray burst][8]][7] [7]:assets/movies/nshe.mov [8]:assets/movies/nshe.png