How Is Angular Resolution Used to Determine the Distance Between Two Stars?

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SUMMARY

The discussion centers on calculating the distance between two stars using angular resolution principles. A radio telescope with a diameter of 20.0 km observes stars 10^7 m away, utilizing a wavelength of 20.0 cm. The minimum resolvable angle is calculated using the formula θ(min) = 1.22 * wavelength / Diameter, yielding a result that leads to a distance of 6.99 x 10^13 m, which is incorrect compared to the expected answer of 1.22 x 10^12 m. The error is attributed to the conversion of angles from radians to degrees, highlighting the importance of using radians for such calculations.

PREREQUISITES
  • Understanding of angular resolution and its significance in astronomy
  • Familiarity with the formula θ(min) = 1.22 * wavelength / Diameter
  • Knowledge of small angle approximation in physics
  • Basic proficiency in unit conversions, particularly between degrees and radians
NEXT STEPS
  • Study the application of the small angle approximation in astrophysics
  • Learn about the principles of radio telescope design and functionality
  • Research the significance of angular resolution in astronomical observations
  • Explore the conversion methods between degrees and radians in mathematical contexts
USEFUL FOR

Astronomy students, astrophysicists, and anyone interested in understanding the principles of angular resolution and its application in measuring astronomical distances.

danmend
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Homework Statement


A radio telescope observes two stars orbiting each other, using radio waves of wavelength 20.0 cm. The stars are 10^7m away from the Earth. The telescope, which has a diameter of 20.0km, can just resolve the two stars. What is the distance between the two stars?

Homework Equations



theta(min) = 1.22* wavelength/Diameter of circular aperture

The Attempt at a Solution



I got 1.22(0.2)/20000 = theta (min). Then I changed it into degrees because I'm more comfortable using them. I got 0.000699 degrees. Then I divided that by half to get the angle to one of the stars. Using the small angle approximation I got 0.000699*10^17m = 3.49x10^13. Then I doubled it to get the distance between the stars 6.99x10^13m. The answer is apparently 1.22x10^12m. Help please :)
 
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Welcome to PF!

Hi danmend ! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)
danmend said:
… Then I changed it into degrees because I'm more comfortable using them.
…
6.99x10^13m. The answer is apparently 1.22x10^12m. Help please :)

6.99x1013 divided by 1.22x1012 = 57.29 = 180/π …

now do you see why it's so much easier to use radians? :smile:​
 

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