AU Lab: Putting It All Together!

In the previous three lab installments, we found the following values:

• 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

The first step in finding the Astronomical Unit is to find the physical size of the sun, which we can find using the sun's rotational period and velocity.  These two quantities are a time and a velocity, which, when multiplied together, provide a distance.  In this case, that distance is the circumference of the sun, which can be used to find the sun's radius.

$P_\odot V_\odot = \boldsymbol{Circ}_\odot =2\pi R_\odot$

$R_\odot = \frac{P_\odot V_\odot}{2\pi}$

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 $\cos \theta =1$