The Hubble Space Telescope (HST) can resolve objects that have a small
angular separation because there is no atmospheric distortion of the light. (i.e. The HST has excellent
resolving power.) Its 2.4-m-diameter primary mirror can collect light from distant galaxies that
formed early in the history of the universe. How far apart can two galaxies be from each other if they
are 10 billion light-years away from Earth and are barely resolved by the HST using visible light with
a wavelength of 400 nm? You may assume that the aperture diameter is the full 2.4 m diameter of the
mirror.
okay so this is what I have tried so far:
I used a sin delta theta>/= 1.22 lambda nought
rearrange to get
delta theta>/= [inverse sin (1.22)(4.00*10^-9m)]/2.4m
so delta theta = 0.00001165
The only reason I used that formula is because it seemed like the best option. I'm not even sure if I'm using the right one.
I don't even know if I'm on the right track, and I have no idea what to do from here. Any help would be appreciated!
You are on the right track but do not appear to be solving for the separation distance of the galaxies from each other. Call that x. That is what they are asking for.
1.22 (lambda)/D = x/R
where R is 10 million light years
lambda = 400*10^-9 m
D = 2.4 m
Solve for x.
If you leave both x and R in light years, you can solve for x in light years.
x/R = 2.03*10^-7 (a dimensionless ratio)
x = 203 light years (when the light was emitted 10 billion light years ago)
My prof says that the answer should be of magnitude 10^19. I'm not sure what he's doing differently. Is there another way to solve this?
I tried to get help from a friend, but her email just said to use tan theta = x/D. I'm not sure why she said to use tan theta, and when I do that-> tan 0.00001165*2.4m=x I get x= 0.000000487 which is obviously not right.
To find out how far apart two galaxies can be from each other if they are barely resolved by the Hubble Space Telescope (HST) using visible light with a wavelength of 400 nm, we need to use the concept of angular resolution.
The angular resolution of a telescope determines its ability to distinguish between two closely spaced objects. In this case, we want to find the angular separation between the two galaxies.
The formula you used, sin(delta theta) >= 1.22 * (wavelength / aperture diameter), is the correct formula for calculating the minimum angular resolution of a telescope. However, to find the actual angular separation between two galaxies, we need to rearrange the formula slightly.
Here's how you can proceed:
1. Rewrite the formula as delta theta >= 1.22 * (wavelength / diameter).
2. Substitute the given values: wavelength = 400 nm (or 400 * 10^-9 m) and diameter = 2.4 m.
3. Calculate the minimum angular separation: delta theta >= 1.22 * (400 * 10^-9 m / 2.4 m).
Simplifying, delta theta >= 1.22 * (2.4 * 10^-7).
Evaluating the right-hand side expression, delta theta >= 2.928 * 10^-7 radians.
4. The angular separation, delta theta, is a measure of how far apart the galaxies appear when seen from Earth. To find the physical separation between the galaxies, we need to multiply the angular separation by their distance from Earth.
Given that the galaxies are 10 billion light-years away, we convert this to meters by using the conversion factor: 1 light-year = 9.461 × 10^15 meters.
Therefore, the distance to the galaxies from Earth = 10 billion years * (9.461 × 10^15 meters/year).
Evaluating this expression, the distance to the galaxies from Earth = 9.461 × 10^25 meters.
5. Finally, to find the actual separation between the two galaxies, multiply the angular separation by the distance:
Separation = delta theta * distance.
Separation = (2.928 * 10^-7 radians) * (9.461 × 10^25 meters).
Evaluating this expression, the actual separation between the two galaxies is approximately 2.77 × 10^19 meters.
So, the two galaxies can be approximately 2.77 × 10^19 meters apart from each other if they are barely resolved by the Hubble Space Telescope using visible light with a wavelength of 400 nm.