Chapter 22, Problem 25Q

To determine

The error in the calculated mass of our galaxy inside the Sun’s orbit and the explanation for that if the calculated distance to the galactic center was off by $10\%$ and if the calculated orbital velocity was off by $10\%.$

Answer to Problem 25Q

The mass of the galaxy if the calculated distance to the galactic center was off by $10\%$ is $0.9\text{times}$ of the original mass of our galaxy inside the Sun’s orbit.

The mass of the galaxy if the calculated orbital velocity was off by $10\%$ is $0.81\text{times}$ of the original mass of our galaxy inside the Sun’s orbit.

Explanation of Solution

Given:

The calculated distance to the galactic center was off by $10\%.$

The calculated orbital velocity was off by $10\%.$

Formula Used:

The mass of the star can be found by using the formula;

$M=\frac{r{\nu }^2}{G}$

Where,

$\begin{array}{l}M=\text{mass of the star's orbit}\\ r=\text{distance from the start to the galactic center}\\ \nu =\text{orbital speed of the star}\\ G=\text{gravitational constant}\end{array}$

Calculation:

Let $M_1$ be the calculated mass of our galaxy inside the Sun’s orbit and $M_2$ be the mass of the galaxy if the calculated distance to the galactic center was off by $10\%.$

$\begin{array}{c}{M}_1=\frac{r{\nu }^2}{G}\mathrm{...}\left(1\right)\\ M_2=\frac{0.9r{\nu }^2}{G}\mathrm{...}\left(2\right)\\ \frac{\left(1\right)}{\left(2\right)}\stackrel{}{\to }\frac{M_1}{M_2}=\frac{1}{0.9}\\ M_2=0.9M_1\end{array}$

And,

Let $M_3$ be the calculated mass of our galaxy inside the Sun’s orbit and $M_4$ be the mass of the galaxy if the calculated orbital velocity was off by $10\%.$

$\begin{array}{c}{M}_3=\frac{r{\nu }^2}{G}\mathrm{...}\left(3\right)\\ M_4=\frac{r{\left(0.9\nu \right)}^2}{G}\mathrm{...}\left(4\right)\\ \frac{\left(3\right)}{\left(4\right)}\stackrel{}{\to }\frac{M_3}{M_4}=\frac{1}{0.81}\\ M_4=0.81M_3\end{array}$

The value of $r$ is off by $10\%$ which causes the mass to be reduced by only $10\%$. However, since the orbital velocity is represented as a squared term in the equation, $10\%$ off of the velocity causes the mass to be decreased more than $10\%$.

Conclusion:

Therefore, the mass of the galaxy if the calculated distance to the galactic center was off by $10\%$ is $0.9\text{times}$ of the original mass of our galaxy inside the Sun’s orbit.

The mass of the galaxy if the calculated orbital velocity was off by $10\%$ is $0.81\text{times}$ of the original mass of our galaxy inside the Sun’s orbit.