We investigate the relation between the star formation rate surface density ( \Sigma _ { SFR } ) and the mass surface density of gas ( \Sigma _ { gas } ) in NGC 5194 ( a.k.a .
M51a , Whirlpool Galaxy ) .
VIRUS-P integral field spectroscopy of the central 4.1 \times 4.1 kpc ^ { 2 } of the galaxy is used to measure H \alpha , H \beta , [ NII ] \lambda \lambda 6548,6584 , and [ SII ] \lambda \lambda 6717,6731 emission line fluxes for 735 regions \sim 170 pc in diameter .
We use the Balmer decrement to calculate nebular dust extinctions , and correct the observed fluxes in order to measure accurately \Sigma _ { SFR } in each region .
Archival HI 21cm and CO maps with similar spatial resolution to that of VIRUS-P are used to measure the atomic and molecular gas surface density for each region .
We present a new method for fitting the Star Formation Law ( SFL ) , which includes the intrinsic scatter in the relation as a free parameter , allows the inclusion of non-detections in both \Sigma _ { gas } and \Sigma _ { SFR } , and is free of the systematics involved in performing linear correlations over incomplete data in logarithmic space .
After rejecting regions whose nebular spectrum is affected by the central AGN in NGC 5194 , we use the [ SII ] /H \alpha ratio to separate spectroscopically the contribution from the diffuse ionized gas ( DIG ) in the galaxy , which has a different temperature and ionization state from those of H II regions in the disk .
The DIG only accounts for 11 % of the total H \alpha luminosity integrated over the whole central region , but on local scales it can account for up to a 100 % of the H \alpha emission , especially in the inter-arm regions .
After removing the DIG contribution from the H \alpha fluxes , we measure a slope N = 0.82 \pm 0.05 , and an intrinsic scatter \epsilon = 0.43 \pm 0.02 dex for the molecular gas SFL .
We also measure a typical depletion timescale \tau = \Sigma _ { HI + H _ { 2 } } / \Sigma _ { SFR } \approx 2 Gyr , in good agreement with recent measurements by ( 3 ) .
The atomic gas density shows no correlation with the SFR , and the total gas SFL in the sampled density range closely follows the molecular gas SFL .
Integral field spectroscopy allows a much cleaner measurement of H \alpha emission line fluxes than narrow-band imaging , since it is free of the systematics introduced by continuum subtraction , underlying photospheric absorption , and contamination by the [ NII ] doublet .
We assess the validity of different corrections usually applied in narrow-band measurements to overcome these issues and find that while systematics are introduced by these corrections , they are only dominant in the low surface brightness regime .
The disagreement with the previous measurement of a super-linear molecular SFL by ( 29 ) is most likely due to differences in the fitting method .
Our results support the recent evidence for a low , and close to constant , star formation efficiency ( SFE= \tau ^ { -1 } ) in the molecular component of the ISM .
The data shows an excellent agreement with the recently proposed model of the SFL by ( 35 ) .
The large intrinsic scatter observed may imply the existence of other parameters , beyond the availability of gas , which are important at setting the SFR .