Answer:
a) How many parsecs are there in one astronomical unit?
![4.85x10^{-6}pc](https://tex.z-dn.net/?f=4.85x10%5E%7B-6%7Dpc)
(b) How many meters are in a parsec?
![3.081x10^{16}m](https://tex.z-dn.net/?f=3.081x10%5E%7B16%7Dm)
(c) How many meters in a light-year?
![9.46x10^{15}m](https://tex.z-dn.net/?f=9.46x10%5E%7B15%7Dm)
(d) How many astronomical units in a light-year?
![63325AU](https://tex.z-dn.net/?f=63325AU)
(e) How many light-years in a parsec?
3.26ly
Explanation:
The parallax angle can be used to find out the distance using triangulation. Making a triangle between the nearby star, the Sun and the Earth, knowing that the distance between the Earth and the Sun (
) is defined as 1 astronomical unit:
![\tan{p} = \frac{1AU}{d}](https://tex.z-dn.net/?f=%5Ctan%7Bp%7D%20%3D%20%5Cfrac%7B1AU%7D%7Bd%7D)
Where d is the distance to the star.
Since p is small it can be represent as:
(1)
Where p(rad) is the value of in radians
However, it is better to express small angles in arcseconds
(2)
Then, equation 2 can be replace in equation 1:
(3)
From equation 3 it can be see that ![1pc = 2.06x10^5 AU](https://tex.z-dn.net/?f=1pc%20%3D%202.06x10%5E5%20AU)
<em>a) How many parsecs are there in one astronomical unit? </em>
⇒
<em>(b) How many meters are in a parsec? </em>
⇒
<em>(c) How many meters in a light-year? </em>
To determine the number of meters in a light-year it is necessary to use the next equation:
![x = c.t](https://tex.z-dn.net/?f=x%20%3D%20c.t)
Where c is the speed of light (
) and x is the distance that light travels in 1 year.
In 1 year they are 31536000 seconds
![x = (3x10^{8}m/s)(31536000s)](https://tex.z-dn.net/?f=x%20%3D%20%283x10%5E%7B8%7Dm%2Fs%29%2831536000s%29)
![x = 9.46x10^{15}m](https://tex.z-dn.net/?f=x%20%3D%209.46x10%5E%7B15%7Dm)
<em>(d) How many astronomical units in a light-year?</em>
⇒
<em>(e) How many light-years in a parsec?</em>
⇒