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Abstract This review summarizes the current state of research aiming at a description of the global heliosphere using both analytical and numerical modeling efforts, particularly in view of the overall plasma/neutral flow and magnetic field structure, and its relation to energetic neutral atoms. Being part of a larger volume on current heliospheric research, it also lays out a number of key concepts and describes several classic, though still relevant early works on the topic. Regarding numerical simulations, emphasis is put on magnetohydrodynamic (MHD), multi-fluid, kinetic-MHD, and hybrid modeling frameworks. Finally, open issues relating to the physical relevance of so-called “croissant” models of the heliosphere, as well as the general (dis)agreement of model predictions with observations are highlighted and critically discussed. 

Abstract MESSENGER observations suggest a magma ocean formed on proto-Mercury, during which evaporation of metals and outgassing of C- and H-bearing volatiles produced an early atmosphere. Atmospheric escape subsequently occurred by plasma heating, photoevaporation, Jeans escape, and photoionization. To quantify atmospheric loss, we combine constraints on the lifetime of surficial melt, melt composition, and atmospheric composition. Consideration of two initial Mercury sizes and four magma ocean compositions determines the atmospheric speciation at a given surface temperature. A coupled interior–atmosphere model determines the cooling rate and therefore the lifetime of surficial melt. Combining the melt lifetime and escape flux calculations provides estimates for the total mass loss from early Mercury. Loss rates by Jeans escape are negligible. Plasma heating and photoionization are limited by homopause diffusion rates of ∼10⁶kg s⁻¹. Loss by photoevaporation depends on the timing of Mercury formation and assumed heating efficiency and ranges from ∼10^6.6to ∼10^9.6kg s⁻¹. The material for photoevaporation is sourced from below the homopause and is therefore energy limited rather than diffusion limited. The timescale for efficient interior–atmosphere chemical exchange is less than 10,000 yr. Therefore, escape processes only account for an equivalent loss of less than 2.3 km of crust (0.3% of Mercury’s mass). Accordingly, ≤0.02% of the total mass of H₂O and Na is lost. Therefore, cumulative loss cannot significantly modify Mercury’s bulk mantle composition during the magma ocean stage. Mercury’s high core:mantle ratio and volatile-rich surface may instead reflect chemical variations in its building blocks resulting from its solar-proximal accretion environment.