  We use new interior models of cold planets to investigate the mass-radius
relationships of solid exoplanets, considering planets made primarily of iron,
silicates, water, and carbon compounds. We find that the mass-radius
relationships for cold terrestrial-mass planets of all compositions we
considered follow a generic functional form that is not a simple power law:
$\log_{10} R_s = k_1 + 1/3 \log_{10}(M_s) - k_2 M_s^{k_3}$ for up to $M_p
\approx 20 M_{\oplus}$, where $M_s$ and $R_s$ are scaled mass and radius
values. This functional form arises because the common building blocks of solid
planets all have equations of state that are well approximated by a modified
polytrope of the form $\rho = \rho_0 + c P^n$.
  We find that highly detailed planet interior models, including temperature
structure and phase changes, are not necessary to derive solid exoplanet bulk
composition from mass and radius measurements. For solid exoplanets with no
substantial atmosphere we have also found that: with 5% fractional uncertainty
in planet mass and radius it is possible to distinguish among planets composed
predominantly of iron or silicates or water ice but not more detailed
compositions; with $\sim$~5% uncertainty water ice planets with $\gtrsim 25%$
water by mass may be identified; the minimum plausible planet size for a given
mass is that of a pure iron planet; and carbon planet mass-radius relationships
overlap with those of silicate and water planets due to similar zero-pressure
densities and equations of state. We propose a definition of "super Earths''
based on the clear distinction in radii between planets with significant gas
envelopes and those without.
