10/04 - 10/08

Measurement Methods

We look at mass as a function of angle $$m(\theta)= 2\pi \rho\ r^2 \sin\theta\ dr\ d\theta$$ and mass per solid angle $$m_{SA} = \frac{m(\theta)}{\cos(\theta+d\theta/2)-\cos(\theta-d\theta/2)}$$ and note four angles which contain

• (blue) Innermost 25% of $m$, $m_{SA}$
• (orange) Innermost 50% of $m$, $m_{SA}$
• (green) Innermost 75% of $m$, $m_{SA}$
• (red) Half the peak value

Since some of these are unfinished, we plot this as a function of radius and determine where to measure. (see color key above for all plots)

Finished profile. Measure from last zone.

Uninished profile. Measure from $\sim$ 110th zone.

Plotting these angles on the profiles, there is not much difference regardless of profile shape (or if $m(\theta)$ vs $m_{SA}$ is used). Vertical lines are innermost 25%, 50%, 75% angles, horizontal line is half-maximum value.

Finished profile: Measure from last zone.

Uninished profile. Measure from $\sim$ 110th zone.

Measurements

Plots for all profiles are shown below, angles are in degrees. (Click to enlarge.)

Using $m(\theta)$

Variable $q$, fixed $v_\text{wind}$ and $v_\text{rto}$.

Variable $v_\text{wind}$, fixed $q$ and $v_\text{rto}$.

Variable $v_\text{rto}$, fixed $q$ and $v_\text{wind}$.

Using $m_{SA}$

Variable $q$, fixed $v_\text{wind}$ and $v_\text{rto}$.

Variable $v_\text{wind}$, fixed $q$ and $v_\text{rto}$.

Variable $v_\text{rto}$, fixed $q$ and $v_\text{wind}$.