  We quantitatively address the conjecture that magnetic helicity must be shed
from the Sun by eruptions launching coronal mass ejections in order to limit
its accumulation in each hemisphere. By varying the ratio of guide and
strapping field and the flux rope twist in a parametric simulation study of
flux rope ejection from approximately marginally stable force-free equilibria,
different ratios of self- and mutual helicity are set and the onset of the
torus or helical kink instability is obtained. The helicity shed is found to
vary over a broad range from a minor to a major part of the initial helicity,
with self helicity being largely or completely shed and mutual helicity, which
makes up the larger part of the initial helicity, being shed only partly.
Torus-unstable configurations with subcritical twist and without a guide field
shed up to about two-thirds of the initial helicity, while a highly twisted,
kink-unstable configuration sheds only about one-quarter. The parametric study
also yields stable force-free flux rope equilibria up to a total
flux-normalized helicity of 0.25, with a ratio of self- to total helicity of
0.32 and a ratio of flux rope to external poloidal flux of 0.94. These results
numerically demonstrate the conjecture of helicity shedding by coronal mass
ejections and provide a first account of its parametric dependence. Both self-
and mutual helicity are shed significantly; this reduces the total initial
helicity by a fraction of $\sim\!0.4\mbox{--}0.65$ for typical source region
parameters.
