A Morse 2-function is a generic smooth map from a manifold M of arbitrary finite dimension to a surface B. Its critical set maps to an immersed collection of cusped arcs in B. The aim of this paper is to explain exactly when it is possible to move th
ese arcs around in B by a homotopy and to give a library of examples when M is a closed 4-manifold. The last two sections give applications to the theory of crown diagrams of smooth 4-manifolds.
We present ALMA observations of a wide binary system in Orion, with projected separation 440 AU, in which we detect submillimeter emission from the protoplanetary disks around each star. Both disks appear moderately massive and have strong line emiss
ion in CO 3-2, HCO+ 4-3, and HCN 3-2. In addition, CS 7-6 is detected in one disk. The line-to-continuum ratios are similar for the two disks in each of the lines. From the resolved velocity gradients across each disk, we constrain the masses of the central stars, and show consistency with optical-infrared spectroscopy, both indicative of a high mass ratio ~9. The small difference between the systemic velocities indicates that the binary orbital plane is close to face-on. The angle between the projected disk rotation axes is very high, ~72 degrees, showing that the system did not form from a single massive disk or a rigidly rotating cloud core. This finding, which adds to related evidence from disk geometries in other systems, protostellar outflows, stellar rotation, and similar recent ALMA results, demonstrates that turbulence or dynamical interactions act on small scales well below that of molecular cores during the early stages of star formation.
We present the results from a large 850 micron survey of the sigma Orionis cluster using the SCUBA-2 camera on the James Clerk Maxwell Telescope. The 0.5-degree diameter circular region we surveyed contains 297 young stellar objects with an age estim
ated at about 3Myr. We detect 9 of these objects, 8 of which have infrared excesses from an inner disc. We also serendipitously detect 3 non-stellar sources at > 5sigma that are likely background submillimetre galaxies. The 9 detected stars have inferred disc masses ranging from 5 to about 17MJup, assuming similar dust properties as Taurus discs and an ISM gas-to-dust ratio of 100. There is a net positive signal toward the positions of the individually undetected infrared excess sources indicating a mean disc mass of 0.5 MJup . Stacking the emission toward those stars without infrared excesses constrains their mean disc mass to less than 0.3MJup, or an equivalent Earth mass in dust. The submillimetre luminosity distribution is significantly different from that in the younger Taurus region, indicating disc mass evolution as star forming regions age and the infrared excess fraction decreases. Submillimeter Array observations reveal CO emission toward 4 sources demonstrating that some, but probably not much, molecular gas remains in these relatively evolved discs. These observations provide new constraints on the dust and gas mass of protoplanetary discs during the giant planet building phase and provide a reference level for future studies of disc evolution.
We present arcsecond resolution mid-infrared and millimeter observations of the center of the young stellar cluster AFGL961 in the Rosette molecular cloud. Within 0.2 pc of each other, we find an early B star embedded in a dense core, a neighboring s
tar of similar luminosity with no millimeter counterpart, a protostar that has cleared out a cavity in the circumcluster envelope, and two massive, dense cores with no infrared counterparts. An outflow emanates from one of these cores, indicating a deeply embedded protostar, but the other is starless, bound, and appears to be collapsing. The diversity of states implies either that protostellar evolution is faster in clusters than in isolation or that clusters form via quasi-static rather than dynamic collapse. The existence of a pre-stellar core at the cluster center shows that that some star formation continues after and in close proximity to massive, ionizing stars.
