  We present a scanning tunneling spectroscopy study on quasiparticle states in
vortex cores in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. The energy of the observed
vortex core states shows an approximately linear scaling with the
superconducting gap in the region just outside the core. This clearly
distinguishes them from conventional localized core states, and is a signature
of the mechanism responsible for their discrete appearance in high-temperature
superconductors. The energy scaling of the vortex core states also suggests a
common nature of vortex cores in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and
YBa$_2$Cu$_3$O$_{7-\delta}$. Finally, the observed vortex core states do not
show any dependence on the applied magnetic field in the range from 1 to 6 T.
