Context : The distance to pulsating stars is classically estimated using the parallax-of-pulsation ( PoP ) method , which combines spectroscopic radial velocity ( RV ) measurements and angular diameter ( AD ) estimates to derive the distance of the star .
A particularly important application of this method is the determination of Cepheid distances in view of the calibration of their distance scale .
However , the conversion of radial to pulsational velocities in the PoP method relies on a poorly calibrated parameter , the projection factor ( p -factor ) .

Aims : We aim to measure empirically the value of the p -factors of a homogeneous sample of nine bright Galactic Cepheids for which trigonometric parallaxes were measured with the Hubble Space Telescope ( HST ) Fine Guidance Sensor by Benedict et al .
( 2007 ) .

Methods : We use the SPIPS algorithm , a robust implementation of the PoP method that combines photometry , interferometry , and radial velocity measurements in a global modeling of the pulsation of the star .
We obtained new interferometric angular diameter measurements using the PIONIER instrument at the Very Large Telescope Interferometer ( VLTI ) , completed by data from the literature .
Using the known distance as an input , we derive the value of the p -factor of the nine stars of our sample and study its dependence with the pulsation period .

Results : We find the following p -factors : p = 1.20 \pm 0.12 for RT Aur , p = 1.48 \pm 0.18 for T Vul , p = 1.14 \pm 0.10 for FF Aql , p = 1.31 \pm 0.19 for Y Sgr , p = 1.39 \pm 0.09 for X Sgr , p = 1.35 \pm 0.13 for W Sgr , p = 1.36 \pm 0.08 for \beta Dor , p = 1.41 \pm 0.10 for \zeta Gem , and p = 1.23 \pm 0.12 for \ell Car .

Conclusions : The values of the p -factors that we obtain are consistently close to p = 1.324 \pm 0.024 .
We observe some dispersion around this average value , but the observed distribution is statistically consistent with a constant value of the p -factor as a function of the pulsation period ( \chi ^ { 2 } = 0.669 ) .
The error budget of our determination of the p -factor values is presently dominated by the uncertainty on the parallax , a limitation that will soon be waived by Gaia .