# 9.5 Count Rates: Slitless Spectroscopy

We now turn to estimation of count rates for slitless spectroscopy using the WFC3 grisms. In this case we are dealing with a dispersed image of the source.
For a point source with a continuous spectrum, the count rate, C, is calculated per pixel along the dispersed spectral trace, and is integrated over a fixed extraction height Nspix in the spatial direction perpendicular to the dispersion. The count rate is:

where:

• Sλ is the total point source sensitivity in units of es−1 per incident erg cm2 s−1 Å1; and $//$
• d is the spectral dispersion in Å pixel−1.
• $//$ is the fraction of the point-source energy within Nspix in the spatial direction.
• the other quantities are as defined in the previous section.

For a source with an unresolved emission line with a flux of in units of erg cm2 s−1, the total count rate recorded over the Nspix extraction height is:

These counts will be distributed over pixels in the dispersion direction according to the instrumental line-spread function.

In contrast to the case of imaging sensitivity Sλ, the spectroscopic point source sensitivity calibration $//$ for a default extraction height of Nspix was measured directly from observations of stellar flux standards during SMOV and Cycle 17 calibration programs 11934, 11936, and 11552. Therefore, the accuracy in laboratory determinations of Tλ for the WFC3 grisms are not crucial to the final accuracy of their sensitivity calibrations.
The peak es−1 pixel−1 from the point source is given by:

where:

• $//$ is the fraction of energy contained within the peak pixel.
• the other quantities are as above.