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The exoplanet field now abounds with new discoveries of planets and planetary systems. It is of great interest for the scientific community to understand the compositions and internal structures of these new planets outside our own solar system, and then infer their formation scenarios. In particular, the proper implementation of the equation-of-states of various matters into the calculation of internal structure is crucial. Thus, we have come up with this write-up that provides the detailed steps to turn an equation-of-state (EOS) into a mass-radius relation of exoplanets, and highlight the physics and chemistry involved in these calculations. It must be visualized!, For pedagogical purpose only (non-profit). Comments and suggestions welcome!

We explain the overall equilibrium-temperature-dependent trend in the exoplanet mass-radius diagram, using the escape mechanisms of hydrogen and relevant volatiles, and the chemical equilibrium calculation of molecular hydrogen (H$_2$) break-up into atomic hydrogen (H). We identify two Cosmic Hydrogen and Ice Loss Lines (CHILLs) in the mass-radius diagram. Gas disks are well known to disperse in ten million years. However, gas-rich planets may lose some or almost all gas on a much longer timescale. We thus hypothesize that most planets that are born out of a hydrogen-gas-dominated nebular disk begin by possessing a primordial H$_2$-envelope. This envelope is gradually lost due to escape processes caused by host-stellar radiation., Comment: submitted to AAS Journals. Comments and suggestions welcome!

Applying the survival function analysis to the planet radius distribution of the Kepler exoplanet candidates, we have identified two natural divisions of planet radius at 4 Earth radii and 10 Earth radii. These divisions place constraints on planet formation and interior structure model. The division at 4 Earth radii separates small exoplanets from large exoplanets above. When combined with the recently-discovered radius gap at 2 Earth radii, it supports the treatment of planets 2-4 Earth radii as a separate group, likely water worlds. Thus, for planets around solar-type FGK main-sequence stars, we argue that 2 Earth radii is the separation between water-poor and water-rich planets, and 4 Earth radii is the separation between gas-poor and gas-rich planets. We confirm that the slope of survival function in between 4 and 10 Earth radii to be shallower compared to either ends, indicating a relative paucity of planets in between 4-10 Earth radii, namely, the sub-Saturnian desert there. We name them transitional planets, as they form a bridge between the gas-poor small planets and gas giants. Accordingly, we propose the following classification scheme: (10 Earth radii) gas giants., 10 pages, 2 figures, under review with MNRAS, comments are very welcome and much appreciated!