  The newly detected TRAPPIST-1 system, with seven low-mass, roughly
Earth-sized planets transiting a nearby ultra-cool dwarf, is one of the most
important exoplanet discoveries to date. The short baseline of the available
discovery observations, however, means that the planetary masses (obtained
through measurement of transit timing variations of the planets of the system)
are not yet well constrained. The masses reported in the discovery paper were
derived using a combination of photometric timing measurements obtained from
the ground and from the Spitzer spacecraft, and have uncertainties ranging from
30\% to nearly 100\%, with the mass of the outermost, $P=18.8\,{\rm d}$, planet
h remaining unmeasured. Here, we present an analysis that supplements the
timing measurements of the discovery paper with 73.6 days of photometry
obtained by the K2 Mission. Our analysis refines the orbital parameters for all
of the planets in the system. We substantially improve the upper bounds on
eccentricity for inner six planets (finding $e<0.02$ for inner six known
members of the system), and we derive masses of $0.79\pm 0.27 \,M_{\oplus}$,
$1.63\pm 0.63\,M_{\oplus}$, $0.33\pm 0.15\,M_{\oplus}$,
$0.24^{+0.56}_{-0.24}\,M_{\oplus}$, $0.36\pm 0.12\,M_{\oplus}$, $0.566\pm
0.038\,M_{\oplus}$, and $0.086\pm 0.084\,M_{\oplus}$ for planets b, c, d, e, f,
g, and h, respectively.
