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SPHEREx is a proposed NASA MIDEX mission selected for Phase A study. SPHERExwould carry out the first all-sky spectral survey in the near infrared. At theend of its two-year mission, SPHEREx would obtain 0.75-to-5$\mu$m spectra ofevery 6.2 arcsec pixel on the sky, with spectral resolution R>35 and a5-$\sigma$ sensitivity AB$>$19 per spectral/spatial resolution element. Moredetails concerning SPHEREx are available at http://spherex.caltech.edu. TheSPHEREx team has proposed three specific science investigations to be carriedout with this unique data set: cosmic inflation, interstellar and circumstellarices, and the extra-galactic background light. Though these three themes areundoubtedly compelling, they are far from exhausting the scientific output ofSPHEREx. Indeed, SPHEREx would create a unique all-sky spectral databaseincluding spectra of very large numbers of astronomical and solar systemtargets, including both extended and diffuse sources. These spectra wouldenable a wide variety of investigations, and the SPHEREx team is dedicated tomaking the data available to the community to enable these investigations,which we refer to as Legacy Science. To that end, we have sponsored twoworkshops for the general scientific community to identify the most interestingLegacy Science themes and to ensure that the SPHEREx data products areresponsive to their needs. In February of 2016, some 50 scientists from allfields met in Pasadena to develop these themes and to understand theirimplications for the SPHEREx mission. The 2016 workshop highlighted manysynergies between SPHEREx and other contemporaneous astronomical missions,facilities, and databases. Consequently, in January 2018 we convened a secondworkshop at the Center for Astrophysics in Cambridge to focus specifically onthese synergies. This white paper reports on the results of the 2018 SPHERExworkshop.

SPHEREx is a proposed SMEX mission selected for Phase A. SPHEREx will carryout the first all-sky spectral survey and provide for every 6.2" pixel aspectra between 0.75 and 4.18 $\mu$m [with R$\sim$41.4] and 4.18 and 5.00$\mu$m [with R$\sim$135]. The SPHEREx team has proposed three specific scienceinvestigations to be carried out with this unique data set: cosmic inflation,interstellar and circumstellar ices, and the extra-galactic background light.It is readily apparent, however, that many other questions in astrophysics andplanetary sciences could be addressed with the SPHEREx data. The SPHEREx teamconvened a community workshop in February 2016, with the intent of enlistingthe aid of a larger group of scientists in defining these questions. This papersummarizes the rich and varied menu of investigations that was laid out. Itincludes studies of the composition of main belt and Trojan/Greek asteroids;mapping the zodiacal light with unprecedented spatial and spectral resolution;identifying and studying very low-metallicity stars; improving stellarparameters in order to better characterize transiting exoplanets; studyingaliphatic and aromatic carbon-bearing molecules in the interstellar medium;mapping star formation rates in nearby galaxies; determining the redshift ofclusters of galaxies; identifying high redshift quasars over the full sky; andproviding a NIR spectrum for most eROSITA X-ray sources. All of theseinvestigations, and others not listed here, can be carried out with the nominalall-sky spectra to be produced by SPHEREx. In addition, the workshop definedenhanced data products and user tools which would facilitate some of thesescientific studies. Finally, the workshop noted the high degrees of synergybetween SPHEREx and a number of other current or forthcoming programs,including JWST, WFIRST, Euclid, GAIA, K2/Kepler, TESS, eROSITA and LSST.