What is the Angular Resolution Formula for the Hubble Space Telescope?

In summary: No worries, it happens to the best of us. Overall, your understanding of the concept was correct and you were able to find the correct formula online. Keep up the good work!In summary, the conversation revolved around finding the smallest object that the Hubble Space Telescope can see on the surface of the Moon and the formula used to derive it. The correct formula was found to be \theta/206265"=d/D, where theta is the angular resolution of the telescope, d is the size of the object on the Moon, and D is the distance between the Earth and the Moon. The conversation also discussed the mistake made in using the formula \theta/360 degrees = distance/2\pir and the correct approach of using
  • #1
colourpalette
11
0
My main issue with this problem is the formula and how to derive it.

Homework Statement


The Hubble Space Telescope has a resolution of about 0.05 arc second. What is the smallest object it could see on the surface of the Moon? Give your answer in metres.


Homework Equations


I tried to get my own equation since there was nothing relevant in my textbook, so I did
[tex]\theta[/tex]/360 degrees = distance/2[tex]\pi[/tex]r
But it's wrong (why?)

I went online and I found
[tex]\theta[/tex]/206265"=d/D
This gives the right answer but I don't understand where it came from.

The Attempt at a Solution


The answer is 93m, I just don't get the formula
 
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  • #2
1 arcsecond = 1/3600 degrees
180 degrees = [itex]\pi[/itex] radians.

for small angles a [tex]\approx tan(a) [/tex] if a is in radians.

You get 93 m if you apply this, but I have no Idea where the number 206265 comes from
 
  • #3
206265 is the number of arcseconds per radian.
 
  • #4
colourpalette said:
I tried to get my own equation since there was nothing relevant in my textbook, so I did
[tex]\theta[/tex]/360 degrees = distance/2[tex]\pi[/tex]r
But it's wrong (why?)

It's wrong because 2*pi*r is the expression for the circumference of a circle. But there is no full circle here, so what are you trying to do?

colourpalette said:
I went online and I found
[tex]\theta[/tex]/206265"=d/D
This gives the right answer but I don't understand where it came from.

remember that the definition of an angle (in radians) is:

theta = s/r​

You can think of an angle as being generated by sweeping out an arc (a portion of a circle) by rotating a line about a fixed point at one of its ends. Here, 's' is the length of the arc that is swept out by the radial line as it rotates by that angle, and 'r' is the radial distance to that arc (in other words it is the length of the line). So, in this case 's' is the size of the surface feature on the moon, and 'r' is the Earth-moon distance. (Alternatively you could use 'd' and 'D' or any other letters you want). You know theta and you know r, so all you have to do is solve this equation for s.

Since the equation is only valid for theta in radians, you have to convert theta to radians by multiplying it by (1/206265) radians/arcsecond.
 
  • #5
cepheid said:
It's wrong because 2*pi*r is the expression for the circumference of a circle. But there is no full circle here, so what are you trying to do?

What I was thinking was the part of the angle (or angular resolution) out of 360 degrees of the whole circle would be the same as arc distance out of the circumference of the circle. And I have the diameter, which is kind of close to the arc distance so my answer would be close.
I know where I went wrong now though
It's out of 2[tex]\pi[/tex], not 360 degrees, thanks for clearing that up!
 
  • #6
Ahh yes I see. Your approach was good, actually! You were thinking that the ratio of the angle subtended to one full rotation should be equal to the ratio of the arc length spanned to one full circle. This is another way of arriving at the same result. You just got tripped up by different unit systems.
 

FAQ: What is the Angular Resolution Formula for the Hubble Space Telescope?

What is the Angular Resolution Problem?

The Angular Resolution Problem, also known as the Rayleigh Criterion, is a limitation in the ability of optical systems to distinguish between two closely spaced objects. It is caused by the diffraction of light, which results in the objects appearing as a single blurred image instead of two distinct images.

How does the Angular Resolution Problem affect telescopes?

The Angular Resolution Problem affects telescopes by limiting their ability to distinguish between closely spaced celestial objects. This can make it difficult to observe and study objects that are close to each other, such as binary stars or galaxies.

Can the Angular Resolution Problem be overcome?

While the Angular Resolution Problem cannot be completely overcome, it can be improved by using larger telescopes or using techniques such as adaptive optics or interferometry. These methods can reduce the effects of diffraction and improve the angular resolution of the telescope.

How is angular resolution measured?

Angular resolution is typically measured in units of arcseconds, which is a unit of angular measurement that is equivalent to 1/3600th of a degree. It is a measure of the smallest angle that can be resolved by a telescope, with smaller values indicating a better angular resolution.

Is the Angular Resolution Problem unique to telescopes?

No, the Angular Resolution Problem is not unique to telescopes. It is a fundamental limitation of all optical systems, including cameras, microscopes, and the human eye. However, the impact of this problem may vary depending on the specific application and the capabilities of the system being used.

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