  We present a detailed investigation of the dramatic changes that occur in the
$\mathcal{L}_1$ halo family when radiation pressure is introduced into the
Sun-Earth circular restricted three-body problem (CRTBP). This
photo-gravitational CRTBP can be used to model the motion of a solar sail
orientated perpendicular to the Sun-line. The problem is then parameterized by
the sail lightness number, the ratio of solar radiation pressure acceleration
to solar gravitational acceleration. Using boundary-value problem numerical
continuation methods and the AUTO software package (Doedel et al. 1991) the
families can be fully mapped out as the parameter $\beta$ is increased.
Interestingly, the emergence of a branch point in the retrograde satellite
family around the Earth at $\beta\approx0.0387$ acts to split the halo family
into two new families. As radiation pressure is further increased one of these
new families subsequently merges with another non-planar family at
$\beta\approx0.289$, resulting in a third new family. The linear stability of
the families changes rapidly at low values of $\beta$, with several small
regions of neutral stability appearing and disappearing. By using existing
methods within AUTO to continue branch points and period-doubling bifurcations,
and deriving a new boundary-value problem formulation to continue the folds and
Krein collisions, we track bifurcations and changes in the linear stability of
the families in the parameter $\beta$ and provide a comprehensive overview of
the halo family in the presence of radiation pressure. The results demonstrate
that even at small values of $\beta$ there is significant difference to the
classical CRTBP, providing opportunity for novel solar sail trajectories.
Further, we also find that the branch points between families in the solar sail
CRTBP provide a simple means of generating certain families in the classical
case.
