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We calculate global $m=1$ modes with low pattern speed corresponding tointroducing a finite eccentricity into a protoplanetary disc. We consider discmodels which are either isolated or contain one or two protoplanets orbiting inan inner cavity. Global modes that are strongly coupled to inner protoplanetsare found to have disc orbits which tend to have apsidal lines antialigned withrespect to those of the inner protoplanets. Other modes corresponding to freedisc modes may be global over a large range of length scales and accordingly belong lived. We consider the motion of a protoplanet in the earth mass rangeembedded in an eccentric disc and determine the equilibrium orbits whichmaintain fixed apsidal alignment with respect to the disc gas orbits.Equilibrium eccentricities are found to be comparable or possibly exceed thedisc eccentricity. We then approximately calculate the tidal interaction withthe disc in order to estimate the orbital migration rate. Results are found todeviate from the case of axisymmetric disc with near circular protoplanet orbitonce eccentricities of protoplanet and disc orbits become comparable to thedisc aspect ratio in magnitude. Aligned protoplanet orbits with very similareccentricity to that of the gas disc are found to undergo litle eccentricitychange while undergoing inward migration in general. However, for significantlylarger orbital eccentricities, migration may be significantly reduced or evenreverse from inwards to outwards. Thus the existence of global non circularmotions in discs with radial excursions comparable to the semi-thickness mayhave important consequences for the migration and survival of protoplanetarycores in the earth mass range.Comment: Accepted for publication by A &

Global $\vec{m} = \mathsf{1}$ modes and migration of protoplanetary cores in eccentric protoplanetary discs