Show that the angular radius of the star is given by....

In summary, the angular radius in arcseconds of a star, as seen from Earth, can be calculated using the formula \theta = (\frac{2.06 \times 10^5}{T_{eff}^2}) \sqrt{\frac{F}{\sigma}}, where T_{eff} is the effective temperature, F is the flux, and \sigma is the Stefan-Boltzmann constant. This can be derived by rearranging equations for luminosity, effective temperature, and flux, and substituting them into each other. The resulting formula involves the ratio of the star's radius to its distance from Earth, which can then be converted to an angle in arcseconds using the factor 2.06 * 10^
  • #1
FaraDazed
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2

Homework Statement


A star has an effective temperature [itex]T_{eff}[/itex] and is observed to have a flux [itex]F[/itex]. Show that the angular radius in arcseconds of the star (as seen from Earth) is given by

[itex]
\theta = (\frac{2.06 \times 10^5}{T_{eff}^2}) \sqrt{\frac{F}{\sigma}}
[/itex]

Homework Equations


[itex]
L = 4 \pi R^2 \sigma T_{eff}^4 \\
T_{eff} = (\frac{L}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\
F = \frac{L}{4 \pi d^2}
[/itex]

And probably..
[tex]
\omega = \frac{A}{R^2}
[/itex] ?

The Attempt at a Solution


[/B]
I am a bit lost here! I missed several of the lecture containing this material, but from looking at what I have go already I should be able to do it?

First I started by rearranging the eqn for Flux for L
[itex]
F = \frac{L}{4 \pi d^2} \\
L = 4 \pi d^2 F
[/itex]
And then subsituted that into the eqn for T_eff
[itex]
T_{eff} = (\frac{L}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\
T_{eff} = (\frac{4 \pi d^2 F}{4 \pi R^2 \sigma})^{\frac{1}{4}} \\
T_{eff}^4 = (\frac{ d^2 F}{R^2 \sigma}) = (\frac{d}{R})^2 (\frac{F}{\sigma}) \\
(\frac{R}{d})^2 = \frac{1}{T_{eff}^4}(\frac{F}{\sigma}) \\
(\frac{R}{d}) = \sqrt{\frac{1}{T_{eff}^4}(\frac{F}{\sigma})} \\
\frac{R}{d} = \frac{1}{T_{eff}^2} \sqrt{\frac{F}{\sigma}} \\
[/itex]

That is as far as I have got. I assume in mine, R/d is the angle in radians, so I hope that it relates to the angle in arcsecond by that 2.06 * 10^5 factor? Or I am way off. I'd appreciate some help/advie with this please.

EDIT: yes It does and I can see how now. My bad.
 
Last edited:
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  • #2
Yep. Looks good.
 
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Related to Show that the angular radius of the star is given by....

1. What is the significance of the angular radius of a star?

The angular radius of a star is an important measurement because it allows us to determine the size of the star and its distance from Earth. This information is crucial for understanding the properties and behavior of the star.

2. How is the angular radius of a star calculated?

The angular radius of a star is calculated using the formula θ = R/D, where θ is the angular radius, R is the physical radius of the star, and D is the distance between the star and Earth. This formula is based on the principles of trigonometry and can be used for stars of any size.

3. Can the angular radius of a star change over time?

Yes, the angular radius of a star can change over time due to factors such as expansion or contraction of the star's outer layers, or changes in its distance from Earth. However, these changes are often very small and difficult to measure.

4. How does the angular radius of a star relate to its luminosity?

There is a direct relationship between the angular radius of a star and its luminosity. Generally, larger stars have larger angular radii and are more luminous, while smaller stars have smaller angular radii and are less luminous. However, this relationship can be affected by other factors such as temperature and composition of the star.

5. Why is it important to measure the angular radius of a star accurately?

Accurate measurements of the angular radius of a star are crucial for understanding its physical properties and behavior. They can also help us determine the star's evolutionary stage, which provides valuable insights into the life cycle of stars and the evolution of our universe.

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