- Rotational Period of the Sun = 28.3 days
- Angular Size of the Sun = 0.56^{o}
- Rotational Velocity of the Sun = 0.975 km/s
P_\odot V_\odot = \boldsymbol{Circ}_\odot =2\pi R_\odot
R_\odot = \frac{P_\odot V_\odot}{2\pi}
Now that we know the radius of the sun, we can use geometry to find the Astronomical Unit, d
where \alpha = \frac{\theta}{2}.
According to the picture above, \tan(\alpha )=\frac{R_\odot}{d}. Because of small angle approximation*, we can say that \tan(\alpha ) = \alpha , which leads us to the equation
\alpha = \frac{\theta}{2}= \frac{R_\odot}{d}
d=\frac{2R_\odot}{\theta}
When you substitute the radius equation for R_\odot, you get this equation for the Astronomical Unit:
d=\frac{\frac{P_\odot V_\odot}{2\pi}}{\frac{\theta}{2}}=\frac{P_\odot V_\odot}{\pi \theta}
Our group found that the distance to the sun is 7.986\times 10^{12} cm, which is about half the actual value. This makes sense because the rotational velocity we found was almost exactly half of the actual value.
*Small angle approximation says that, for \theta < < 1^{o} , \sin \theta = \theta and \cos \theta =1
Hi Moiya,
ReplyDeleteYou covered the basics here, but it would be helpful to see a more detailed discussion of the result. Why do we care about measuring the AU? How would the measurement be improved?