Answer:
a) How many parsecs are there in one astronomical unit?
[tex]4.85x10^{-6}pc[/tex]
(b) How many meters are in a parsec?
[tex]3.081x10^{16}m[/tex]
(c) How many meters in a light-year?
[tex]9.46x10^{15}m[/tex]
(d) How many astronomical units in a light-year?
[tex]63325AU[/tex]
(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 ([tex]1.496x10^{11} m[/tex]) is defined as 1 astronomical unit:
[tex]\tan{p} = \frac{1AU}{d}[/tex]
Where d is the distance to the star.
Since p is small it can be represent as:
[tex]p(rad) = \frac{1AU}{d}[/tex] (1)
Where p(rad) is the value of in radians
However, it is better to express small angles in arcseconds
[tex]p('') = p(rad)\frac{180^\circ}{\pi rad}.\frac{60'}{1^\circ}.\frac{60''}{1'}[/tex]
[tex]p('') = 2.06x10^5 p(rad)[/tex]
[tex]p(rad) = \frac{p('')}{2.06x10^5}[/tex] (2)
Then, equation 2 can be replace in equation 1:
[tex]\frac{p('')}{2.06x10^5} = \frac{1AU}{d}[/tex]
[tex]\frac{d}{1AU} = \frac{2.06x10^5}{p('')}[/tex] (3)
From equation 3 it can be see that [tex]1pc = 2.06x10^5 AU[/tex]
a) How many parsecs are there in one astronomical unit?
[tex]1AU . \frac{1pc}{2.06x10^5AU}[/tex] ⇒ [tex]4.85x10^{-6}pc[/tex]
(b) How many meters are in a parsec?
[tex]2.06x10^{5}AU . \frac{1.496x10^{11}m}{1AU}[/tex] ⇒ [tex]3.081x10^{16}m[/tex]
(c) How many meters in a light-year?
To determine the number of meters in a light-year it is necessary to use the next equation:
[tex]x = c.t[/tex]
Where c is the speed of light ([tex]c = 3x10^{8}m/s[/tex]) and x is the distance that light travels in 1 year.
In 1 year they are 31536000 seconds
[tex]x = (3x10^{8}m/s)(31536000s)[/tex]
[tex]x = 9.46x10^{15}m[/tex]
(d) How many astronomical units in a light-year?
[tex]9.46x10^{15}m . \frac{1AU}{1.496x10^{11}m}[/tex] ⇒ [tex]63325AU[/tex]
(e) How many light-years in a parsec?
[tex]2.06x10^{5}AU . \frac{1ly}{63235AU}[/tex] ⇒ [tex]3.26ly[/tex]
