  Here, we develop scaling laws for (1) the distribution of impact-induced heat
within the mantle and (2) shape of the impact-induced melt based on more than
100 smoothed particle hydrodynamic (SPH) simulations. We use Legendre
polynomials to describe these scaling laws and determine their coefficients by
linear regression, minimizing the error between our model and SPH simulations.
The input parameters are the impact angle $\theta$ ($0^{\circ}, 30^{\circ},
60^{\circ}$, and $90^{\circ}$), total mass $M_T$ ($1M_{\rm Mars}-53M_{\rm
Mars}$, where $M_{\rm Mars}$ is the mass of Mars), impact velocity $v_{\rm
imp}$ ($v_{\rm esc} - 2v_{\rm esc}$, where $v_{\rm esc}$ is the mutual escape
velocity), and impactor-to-total mass ratio $\gamma$ ($0.03-0.5$). We find that
the equilibrium pressure at the base of a melt pool can be higher (up to
$\approx 80 \%$) than those of radially-uniform global magma ocean models. This
could have a significant impact on element partitioning. These melt scaling
laws are publicly available on GitHub
($\href{https://github.com/mikinakajima/MeltScalingLaw}{https://github.com/mikinakajima/MeltScalingLaw}$).
