Imagine that you are an astronomer, and you are planning to do a survey to find new brown-dwarf stars. You need to calculate how sensitive your search needs to be (and hence how big a telescope you will need and how long the survey will take).
You want to find ten times more brown dwarfs than the previous best survey. This previous survey was sensitive to brown dwarfs with fluxes of 4.0e-16 .
In order to find ten times more brown dwarfs, your survey will need to find all brown dwarfs down to what limiting flux? (in )
You may assume that both the old and new surveys cover 100 square degrees of the sky, but that there is no overlap between the areas covered.
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To find the limiting flux needed to detect ten times more brown dwarfs, we need to calculate the increase in sensitivity compared to the previous survey.
1. Determine the number of brown dwarfs detected in the previous survey:
Let's assume the previous survey found X brown dwarfs.
2. Determine the desired number of brown dwarfs to be found in the new survey:
The new survey aims to find ten times more brown dwarfs than the previous survey, so the desired number of brown dwarfs is 10X.
3. Calculate the increase in sensitivity:
To find ten times more brown dwarfs, the new survey needs to be ten times more sensitive than the previous survey. Therefore, we need to find the flux level that is one-tenth of the previous survey's flux.
4. Calculate the limiting flux for the new survey:
The limiting flux for the new survey will be the previous survey's limiting flux divided by ten.
Therefore, the limiting flux for the new survey is (4.0e-16) / 10 = 4.0e-17.
To find the limiting flux, we need to first understand what it represents. Flux is the amount of energy received per unit area per unit time from a celestial object. In this case, we're interested in measuring the flux emitted by brown dwarf stars.
Given that the previous survey was sensitive to brown dwarfs with fluxes of 4.0e-16 and we want to find ten times more brown dwarfs, we can calculate the desired limiting flux.
To do this, we first determine the number of brown dwarf stars detected in the previous survey. Since we're assuming that both surveys cover 100 square degrees of the sky and there is no overlap, we can use the number density of the brown dwarfs to calculate the number of brown dwarfs detected.
Let's assume the number density of brown dwarfs in the previous survey was n dwarfs per square degree. Then, the number of brown dwarfs found in the previous survey is given by:
Number of brown dwarfs = n dwarfs per square degree * 100 square degrees
Now, since we want to find ten times more brown dwarfs than the previous survey, the desired number of brown dwarfs in the new survey is given by:
Desired number of brown dwarfs = 10 * (n dwarfs per square degree * 100 square degrees)
Now, since both surveys cover the same area, the sensitivity of the new survey needs to be adjusted to find the desired number of brown dwarfs. This adjustment can be determined by calculating the limiting flux for the new survey.
The limiting flux corresponds to the faintest brown dwarf that can be detected by the sensitive survey. In other words, any brown dwarf with a flux lower than the limiting flux will not be detected.
To find the limiting flux, we'll use the following equation relating the number of detected objects to the flux:
Number of objects = A * n * ∫(F/F0)^(-3/2) dF
where A is the sky area covered, n is the number density of objects, F is the flux, and F0 is the limiting flux.
Since we know the number of objects for both the previous and new surveys and the relevant parameters for both surveys are the same, we can set up the following equation:
(n * 100 * A) / (n * ∫(F/F0)^(-3/2) dF) (previous survey) = (10 * n * 100 * A) / (n * ∫(F/F0)^(-3/2) dF) (new survey)
Simplifying and canceling the common terms, we get:
∫(F/F0)^(-3/2) dF (previous survey) = 10 * ∫(F/F0)^(-3/2) dF (new survey)
This equation shows that the integral of the flux ratio (F/F0)^(-3/2) for the previous survey is equal to ten times the integral of the flux ratio (F/F0)^(-3/2) for the new survey.
Now, the integral ∫(F/F0)^(-3/2) dF is a well-known mathematical expression that evaluates to a constant value, which is 2. Therefore, we can rewrite the equation as:
2 (previous survey) = 10 * 2 (new survey)
Simplifying further, we get:
(previous survey) = 5 * (new survey)
Since the flux ratio term cancels out, we can conclude that the sensitivity of the new survey needs to be five times better than the previous survey.
Therefore, the limiting flux for the new survey would be 4.0e-16 / 5 = 8.0e-17.