How can we construct a nearby search to immediately get relatively nearby
SNe to augment or replace the Hamuy SNe?


Desirable redshifts:

B --> V    z ~ 0.25 --> m(R)peak ~ 21.0
V --> R    z ~ 0.16 --> m(R)peak ~ 20.0
V --> V    z ~ 0.10 --> m(R)peak ~ 19.0


N(SN) = (0.82 / 100 yr / 10^10 Lsun)    ; rate from Pain et al
      * 15 days / 365 days/yr           ; window for SNe w/ t < tmax (z depend)
      * (2 * 10^8 Lsun h Mpc^-3)        ; luminosity density
      *  50 sq deg / (57.3 rad / deg)^2 ; solid angle
      * 1/3 * (c*zmax/100 h)^3          ; volume per unit solid angle

For zmax = 0.25, this gives 2.1e6 Mpc^3 h^-3 * 6.7e-6 SNe Mpc^3 h = 14.3 h^-2

h = 0.7 --> 29 SNe @ z < 0.25 @ t < tmax over  50 sq deg. --> 0.586 / sq deg
            29 SNe @ z < 0.16 @ t < tmax over 200 sq deg. --> 0.147 / sq deg
            29 SNe @ z < 0.10 @ t < tmax over 800 sq deg. --> 0.037 / sq deg

Peter and Saul have used our SNe Ia rates between 0.3 and 0.6 to estimate that
the rate out to z = 0.1 should be ~ 0.02 SNe / sq deg. This is about 2x lower
than the estimate used above. Mark Phillips estimates from his data that the
rate out to z = 0.1 would be ~ 0.01 SNe / sq deg. So, there is a factor of 4
uncertainty in the local rates. Since the Pain et al rate is very uncertain,
(by 75%), it seems safest to use a rate of ~ 0.02 SNe / sq deg out to z = 0.1.


Follow-up resources:

The follow-up time goes as the inverse square of the average source
brightness, and therefore as the 4th power of the limiting distance!
So, a search aiming for z ~ 0.25 requires 39 times as much time for
equal quality follow-up as one aiming for z ~ 0.1. This is in the
sky-limited case. A z ~ 0.25 search for SN Ia would be sky-limited,
while a search to z ~ 0.1 might have about equal noise contributions
from object and sky (except that sky will again dominate once the
moon is up).

A straight-forward calculation shows that for a volume-limited sample of
standard candles, the mean follow-up time equals 3/7 of the time needed to
observe the most distant object.

 L    = standard candle luminosity
 r    = distance
 rmax = limiting distance
 f    = source flux at    r    = L/(4*pi*r^2)
 fmax = flux of source at rmax = L/(4*pi*rmax^2)
 t    = exposure time 
 tmax = exposure time for source a rmax

 t    ~ f^-2    ~ r^4
 tmax ~ fmax^-2 ~ rmax^4

 <t>      = integral[0,rmax] (t * r^2 dr)          / integral[0,rmax] (r^2 dr)

 <t/tmax> = integral[0,rmax] (t/tmax * r^2 dr)     / integral[0,rmax] (r^2 dr)

 <t/tmax> = integral[0,rmax] (f^2/fmax^2 * r^2 dr) / integral[0,rmax] (r^2 dr)

 <t/tmax> = integral[0,rmax] (r^4/rmax^4 * r^2 dr) / integral[0,rmax] (r^2 dr)

 <t/tmax> = (1/7)*rmax^7/(rmax^4) /((1/3)*rmax^3)

 <t/tmax> = 3/7