To solve this problem, we can use the formula for signal-to-noise ratio (SNR):
SNR = (Signal / Noise)
We are given that the SNR per nanometer is 25. The signal is the number of photons received from the galaxy within the V-filter bandpass, and the noise is the faint glow emitted by the atmosphere.
First, let's calculate the signal:
We know that the magnitude difference between the galaxy and the faint glow is 20 - (-21) = 41 magnitudes. The magnitude system is logarithmic, so the flux ratio between the galaxy and the faint glow is given by:
Flux ratio = 10^(-0.4 * (V_star - V_galaxy))
Since V_star is -21 and V_galaxy is 20, the flux ratio is:
Flux ratio = 10^(-0.4 * (-21 - 20))
= 10^(-0.4 * (-41))
= 10^(16.4)
≈ 2.5 * 10^16
The total signal received from the galaxy within the V-filter bandpass is directly proportional to the flux ratio. Let's assume there are N photons received from the galaxy within the V-filter bandpass:
N ∝ Flux ratio
N = k * Flux ratio
We can solve for k by using the fact that the V-filter has a standard magnitude zero point, which is roughly 3640 photons/cm^2/s/Angstrom. This means that if a star of magnitude 0 is observed through the V-filter, it would produce 3640 photons/cm^2/s/Angstrom:
k * Flux ratio = 3640
Substituting the value of Flux ratio, we get:
k * (2.5 * 10^16) = 3640
k ≈ 3640 / (2.5 * 10^16)
≈ 1.45 * 10^(-13)
Now, we can calculate the signal received from the galaxy within the V-filter bandpass:
Signal = k * Flux ratio
≈ (1.45 * 10^(-13)) * (2.5 * 10^16)
≈ 3.63
Next, let's calculate the noise:
The noise emitted by the atmosphere is equivalent to a V-band magnitude of -21. Assuming the same V-filter zero point, the noise would produce the following number of photons:
Noise = -21 * 3640
≈ -76440
The noise is negative because it represents a faint glow, but we will take its absolute value as it does not affect the calculation of the SNR.
Finally, we can calculate the SNR:
SNR = Signal / Noise
= 3.63 / |(-76440)|
≈ 4.76 * 10^(-5)
To obtain a reliable spectrum, astronomers generally aim for an SNR of at least 10. Let's assume we want an SNR of 10:
SNR = 10 = Signal / Noise
Signal = 10 * Noise
= 10 * |(-76440)|
= 764400
Now, let's calculate the time required to collect enough photons to achieve this signal:
Assuming the telescope collects all the photons from the galaxy within the V-filter bandpass, the number of photons collected per second can be calculated using the formula:
Number of photons per second = Telescope collecting area * Signal
The collecting area of the 8.2m telescope can be calculated using the formula for the area of a circle:
Area = π * (Radius)^2
Given a telescope diameter of 8.2m, the radius can be calculated as:
Radius = Diameter / 2
= 8.2 / 2
= 4.1m
Substituting the value of the radius, we can calculate the collecting area:
Area = π * (4.1)^2
≈ 53.14 m^2
Now, let's calculate the number of photons collected per second:
Number of photons per second = 53.14 * 764400
Assuming the photons are collected evenly across the V-filter bandpass, the number of photons collected per nanometer can be calculated using the formula:
Number of photons per nanometer = Number of photons per second / Bandpass
The V-filter bandpass is typically around 90 nm. Therefore:
Number of photons per nanometer = (53.14 * 764400) / 90
Finally, we can calculate the time required to obtain the optical spectrum by finding the reciprocal of the number of photons collected per nanometer:
Time = 1 / (Number of photons per nanometer)
Solving this equation will give us the desired time in seconds:
Time = 1 / [(53.14 * 764400) / 90]
Calculating this equation will give us the final answer for the time required to obtain the optical spectrum.
Imaging from the Earth's surface causes a problem when even on a clear night the atmosphere emits a faint glow, equivalent to a V
21 star everywhere you look.On such a night, how long would you need to observe on the a 8.2m telescope to obtain an optical spectrum of a V
20 magnitude elliptical galaxy where the signal to noise ratio per nanometer is 25 ?Assume the spectrograph spreads the photons received evenly across the V-filter bandpass. SOLVE IT AND GIVE ANSWERS
1 answer