Math Problem Statement
An astronomer is observing a binary star that is 35 pc away. The two stars have an orbital separation of 5.5 AU. If the astronomer is observing the binary star at a wavelength of 580 nm, what is the minimum aperture of the telescope that is needed in order to resolve the two stars?
Solution
To resolve the two stars in the binary system, we can use the Rayleigh criterion, which gives the minimum angular separation that can be resolved by a telescope. The Rayleigh criterion is given by:
where:
- is the minimum angular separation (in radians),
- is the wavelength of observation (in meters),
- is the aperture of the telescope (in meters).
Step 1: Angular Separation of the Binary Stars
First, we need to calculate the angular separation between the two stars in radians. The orbital separation is given as 5.5 AU, and the distance to the binary system is 35 parsecs (pc). Using the small-angle approximation:
where:
- is the separation between the stars,
- is the distance to the system.
We need to express both and in the same units. Recall:
- ,
- .
So:
Now, calculate the angular separation :
Step 2: Aperture of the Telescope
The astronomer is observing at a wavelength of .
From the Rayleigh criterion: To resolve the stars, , so:
Substitute the values of and :
Final Answer:
The minimum aperture of the telescope needed to resolve the two stars is approximately 0.93 meters.
Would you like more details on this? Here are five related questions you might be interested in:
- How does the Rayleigh criterion change for different wavelengths of light?
- What is the effect of atmospheric turbulence on resolving binary stars?
- How does the aperture of the telescope affect its resolution power?
- How do astronomers measure the angular separation between stars in practice?
- What other factors, besides aperture, influence a telescope's ability to resolve objects?
Tip: Increasing the aperture of a telescope improves its resolving power and allows it to capture finer details in distant objects.
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Math Problem Analysis
Mathematical Concepts
Angular Separation
Rayleigh Criterion
Small Angle Approximation
Astronomical Units
Parsecs
Formulas
Rayleigh criterion: θ_min = 1.22 * λ / D
Angular separation: θ = s / d
Theorems
Rayleigh Criterion
Suitable Grade Level
Undergraduate Astronomy/Physics
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