  We present the results of a spectroscopic survey to characterize
chromospheric activity, as measured by H$\alpha$ emission, in low-mass members
of the 500-Myr-old open cluster M37. Combining our new measurements of
H$\alpha$ luminosities ($L_{H\alpha}$) with previously cataloged stellar
properties, we identify saturated and unsaturated regimes in the dependence of
the $L_{H\alpha}$-to-bolometric-luminosity ratio, $L_{H\alpha}/L_{bol}$, on the
Rossby number $R_o$. All rotators with Ro smaller than 0.03$\pm$0.01 converge
to an activity level of $L_{H\alpha}/L_{bol}$ = (1.27$\pm$0.02) x 10$^{-4}$.
This saturation threshold ($R_{o,sat}$ = 0.03$\pm$0.01) is statistically
smaller than that found in most studies of the rotation-activity relation. In
the unsaturated regime, slower rotators have lower levels of chromospheric
activity, with $L_{H\alpha}/L_{bol}$($R_o$) following a power-law of index
$\beta$ = -0.51$\pm$0.02, slightly shallower than the one found for a combined
$\approx$650-Myr-old sample of Hyades and Praesepe stars. By comparing this
unsaturated behavior to that previously found for coronal activity in M37 (as
measured via the X-ray luminosity, $L_X$), we confirm that chromospheric
activity decays at a much slower rate than coronal activity with increasing
$R_o$. While a comparison of $L_{H\alpha}$ and $L_X$ for M37 members with
measurements of both reveals a nearly 1:1 relation, removing the
mass-dependencies by comparing instead $L_{H\alpha}/L_{bol}$ and $L_X/L_{bol}$
does not provide clear evidence for such a relation. Finally, we find that
$R_{o,sat}$ is smaller for our chromospheric than for our coronal indicator of
activity ($R_{o,sat}$ = 0.03$\pm$0.01 versus 0.09$\pm$0.01). We interpret this
as possible evidence for coronal stripping.

Solar and Stellar Astrophysics