A galaxy with an apparent magnitude in the V band (λc ~ 5550 angstroms, width δλ ~ 800 angstroms) of mV = 22.5 mag is observed with the Palomar 200-inch telescope. How many photons per second are collected from this galaxy over the 200-inch (~ 500 cm) diameter mirror? (assume for simplicity that the filter throughput is = 1 within that bandpass, and that galaxy's photon spectum is flat, and ignore any absorpton or inefficiencies). Answers within roughly 60 photons of the correct answer will be accepted; this question is worth two points.
1.3378e19
150 photons
150
To calculate the number of photons per second collected from the galaxy, we need to use the magnitude system and the properties of the telescope.
The magnitude system is logarithmic, and the formula to convert magnitudes to flux is:
F = F0 * 10^(-0.4*m)
Where:
F is the flux in erg/s/cm^2/angstrom
F0 is the reference flux at magnitude zero (F0 = 3.63078x10^(-9) erg/s/cm^2/angstrom for V band)
First, let's calculate the flux received from the galaxy using the magnitude:
F = F0 * 10^(-0.4*mV)
F = (3.63078x10^(-9) erg/s/cm^2/angstrom) * 10^(-0.4*22.5)
F = (3.63078x10^(-9) erg/s/cm^2/angstrom) * 10^(-9)
F ≈ 2.332x10^(-34) erg/s/cm^2/angstrom
Now, let's calculate the total flux collected by the telescope. We need to consider the telescope's effective area, which is determined by its diameter.
The area of a circle is given by the formula A = πr^2. Since the diameter is given, we have:
A = π * (500/2)^2 cm^2
A = π * 250^2 cm^2
A ≈ 196349.5 cm^2
To calculate the total flux collected by the telescope, we multiply the flux by the effective area:
Total Flux = F * A
Total Flux = (2.332x10^(-34) erg/s/cm^2/angstrom) * 196349.5 cm^2
Total Flux ≈ 4.580x10^(-29) erg/s/angstrom
Finally, to convert this flux to photons per second, we need to divide by the energy per photon.
The energy per photon is given by E = hc/λ, where h is Planck's constant and c is the speed of light. We will use the given wavelength of the V band (λc ~ 5550 angstroms) to calculate it.
E = (6.63x10^(-27) erg*s * 3x10^10 cm/s) / (5550 angstroms * 1x10^(-8) cm/angstrom)
E ≈ 3.784x10^(-16) erg
To calculate the number of photons per second, we divide the total flux by the energy per photon:
Number of photons per second = Total Flux / E
Number of photons per second ≈ (4.580x10^(-29) erg/s/angstrom) / (3.784x10^(-16) erg)
Number of photons per second ≈ 1.209x10^(-13) photons/s
So, the number of photons per second collected from the galaxy over the 200-inch diameter mirror is approximately 1.209x10^(-13) photons/s.