Far-infrared/sub-millimetre properties of pre-stellar cores L1521E,
  L1521F and L1689B as revealed by the Herschel SPIRE instrument -- I. Central
  positions
  Dust grains play a key role in the physics of star-forming regions, even
though they constitute only $\sim$1 % of the mass of the interstellar medium.
The derivation of accurate dust parameters such as temperature ($T_{dust}$),
emissivity spectral index ($\beta$) and column density requires broadband
continuum observations at far-infrared wavelengths. We present Herschel-SPIRE
Fourier Transform Spectrometer (FTS) measurements of three starless cores:
L1521E, L1521F and L1689B, covering wavelengths between 194 and 671 $\mu$m.
This paper is the first to use our recently updated SPIRE-FTS intensity
calibration, yielding a direct match with SPIRE photometer measurements of
extended sources. In addition, we carefully assess the validity of calibration
schemes depending on source extent and on the strength of background emission.
The broadband far-infrared spectra for all three sources peak near 250 $\mu$m.
Our observations therefore provide much tighter constraints on the spectral
energy distribution (SED) shape than measurements that do not probe the SED
peak. The spectra are fitted using modified blackbody functions, allowing both
$T_{dust}$ and $\beta$ to vary as free parameters. This yields $T_{dust}$ of
9.8$\pm$0.2 K, 15.6$\pm$0.5 K and 10.9$\pm$0.2 K and corresponding $\beta$ of
2.6$\mp$0.9, 0.8$\mp$0.1 and 2.4$\mp$0.8 for L1521E, L1521F and L1689B
respectively. The derived core masses are 1.0$\pm$0.1, 0.10$\pm$0.01 and
0.49$\pm$0.05 $M_{\odot}$, respectively. The core mass/Jeans mass ratios for
L1521E and L1689B exceed unity indicating that they are unstable to
gravitational collapse, and thus pre-stellar cores. By comparison, the elevated
temperature and gravitational stability of L1521F support previous arguments
that this source is more evolved and likely a protostar.
