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.
To determine the required observation time, we need to consider the signal-to-noise ratio (SNR) and the exposure time.
The SNR is given by the formula:
SNR = (flux of the object) / (sqrt(flux of the sky background) + sqrt(flux due to read noise))
In this case, the flux of the object (galaxy) is given by the magnitude difference:
flux of the object = 10^((V_mag - V_star)/2.5)
where V_mag is the magnitude of the galaxy and V_star is the magnitude of the reference star.
The flux of the sky background is given by the magnitude of the sky glow:
flux of the sky background = 10^((V_sky - V_star)/2.5)
where V_sky is the magnitude of the sky background.
Assuming the spectrograph spreads the photons received evenly across the V-filter bandpass, the total flux for the galaxy signal would be distributed across the V-filter bandpass.
Given that the SNR per nanometer is 25, we can use this information to obtain the exposure time required to achieve the desired SNR. Let's assume the V-filter bandpass is 100 nm wide.
SNR = (flux of the object per nanometer) / (sqrt(flux of the sky background per nanometer) + sqrt(flux due to read noise))
The flux of the object per nanometer is given by:
flux of the object per nanometer = (flux of the object) / 100
The flux of the sky background per nanometer is given by:
flux of the sky background per nanometer = (flux of the sky background) / 100
Rearranging the SNR equation, we get:
SNR = (flux of the object per nanometer) / (sqrt(flux of the sky background per nanometer) + sqrt(flux due to read noise))
Solving for the exposure time (t):
t = (SNR^2 * (flux of the sky background per nanometer) + SNR * sqrt(flux of the sky background per nanometer)) / flux of the object per nanometer
Substituting the magnitude values and calculating the flux values using the aforementioned formulas, we can find the required observation time.
To calculate the time needed to observe an optical spectrum of a magnitude 20 elliptical galaxy using an 8.2m telescope with a signal-to-noise ratio (SNR) of 25 per nanometer, we can follow these steps:
Step 1: Calculate the number of photons received per second from the galaxy.
- The V-band magnitude difference between the faintest visible star and the elliptical galaxy is 21 - 20 = 1 magnitude.
- The magnitude scale is logarithmic, so a 1 magnitude difference corresponds to a flux difference of 2.512 times.
- Since the SNR is per nanometer, we need to convert the V-band magnitude into flux per nanometer.
- The V-band flux per nanometer is equivalent to the flux of a V=0 star, which is 3.6 x 10^9 photons per second per square centimeter per nanometer.
Step 2: Calculate the number of photons received from the galaxy through the telescope.
- The collecting area of an 8.2m telescope is A = π * (8.2/2)^2 = 53.19 m^2 = 5.319 x 10^10 cm^2.
- Assuming the spectrograph spreads the photons received evenly across the V-filter bandpass, the effective collecting area is reduced by the bandwidth of the V filter.
- The V-filter bandpass is typically around 400-700 nm, so we can assume a width of 300 nm.
- The effective collecting area is then reduced by a factor of 300 nm / (400-700 nm) = 0.429.
Step 3: Calculate the number of photons per second received from the galaxy through the telescope.
- Multiply the flux per nanometer by the effective collecting area:
Number of photons received per second = (3.6 x 10^9 photons/cm^2/s/nm) * (5.319 x 10^10 cm^2) * 0.429
Step 4: Calculate the number of seconds needed to obtain a desired signal-to-noise ratio (SNR).
- The SNR is given as 25 per nanometer, so the total SNR over the V-band (300 nm) would be 25 * 300 = 7500.
- Divide the desired SNR by the actual SNR:
Time needed (seconds) = Total SNR / SNR per second = 7500 / (Number of photons received per second)
By following these steps and plugging in the values, you can calculate the time needed to observe the optical spectrum of a magnitude 20 elliptical galaxy with the given specifications.
To determine how long you would need to observe on an 8.2m telescope to obtain an optical spectrum of a V
20 magnitude elliptical galaxy with a signal-to-noise ratio of 25, you need to consider various factors.
The first step is to understand the relationship between exposure time, signal-to-noise ratio (SNR), and magnitude. The SNR is given by the formula:
SNR = sqrt(signal) / sqrt(signal + background + noise)
Here, the signal corresponds to the number of photons captured from the galaxy, the background is the contribution from the faint glow of the Earth's atmosphere, and the noise represents the random fluctuations in the measurements.
Since the signal from the galaxy is directly related to its magnitude, you can use the following equation:
signal = 10^(-0.4 * (V_mag - zero_point_mag))
In this equation, V_mag is the magnitude of the galaxy (V
20 in this case), and zero_point_mag is a reference magnitude that corresponds to a known flux or certain conditions. For V
-band observations, the commonly used zero-point magnitude is approximately V
= 0.
To calculate the background contribution from the Earth's atmosphere, you need to determine the number of photons per square nanometer per square meter per second. Let's assume this background count rate is b photons/(nm m² s).
Now, to calculate the total background signal, you need to consider the width of the V
-filter bandpass. Let's assume it is B nm. The total background signal (background) is given by:
background = b * B * t
where t is the exposure time in seconds.
To achieve the desired signal-to-noise ratio (SNR = 25), you can rearrange the SNR equation to solve for the exposure time:
t = (SNR^2 * (signal + background + noise)) / b
Again, the noise term represents additional sources of noise like readout noise, instrumental noise, etc., which are beyond the scope of this explanation.
By plugging in the values you have, you can compute the exposure time required to achieve the desired SNR.
It's important to note that this calculation assumes a simplified scenario and neglects other factors like telescope efficiency, atmospheric conditions, and instrumental noise. These factors can have a significant impact on the final exposure time.
Overall, using the formulae mentioned above, you can calculate the exposure time needed for your specific requirements of observing the V
20 magnitude elliptical galaxy with a signal-to-noise ratio of 25 using an 8.2m telescope.