Luminosity, Flux, Time: What Do They Mean?

[StephenPrivitera]

L=A[sig]T⁴
f=L/A=[sig]T⁴
Where does time come into these equations?

If one telescope of a known diameter can reach a certain magnitude, it is easy to find what magnitude a different telescope of known magnitude can reach. But if a telescope integrating over 10s can reach a certain magnitude, how can we determine what magnitude the same telescope will reach integrating over 1hr?

Also, what does it mean for something to have a spectral resolution of 5?

 

[Nereid]

Where does time come into these equations?

What are the units of [sig]?

But if a telescope integrating over 10s can reach a certain magnitude, how can we determine what magnitude the same telescope will reach integrating over 1hr?

Assuming you're still way above sky noise, presumably.
If one star has a magnitude of 5, and another 10, what is the ratio of the energy received, in unit time, by the same telescope?
How many seconds are there in an hour?