Question

The astronomical unit (AU) is defined as the mean center-to-center distance from Earth to the Sun, namely 1.496x10^(11) m. The parsec is the radius of a circle for which a central angle of 1 s intercepts an arc of length 1 AU. The light-year is the distance that light travels in 1 y.

(a) How many parsecs are there in one astronomical unit?
(b) How many meters are in a parsec?
(c) How many meters in a light-year? (d) How many astronomical units in a light-year? (e) How many light-years in a parsec?

Answer 1

a) How many parsecs are there in one astronomical unit?

`4.85x10^(-6)pc`

(b) How many meters are in a parsec?

`3.081x10^(16)m`

(c) How many meters in a light-year?

`9.46x10^(15)m`

(d) How many astronomical units in a light-year?

`63325AU`

(e) How many light-years in a parsec?

3.26ly

Step-by-step 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 (
`1.496x10^(11) m`) is defined as 1 astronomical unit:

`tan(p) = (1AU)/(d)`

Where d is the distance to the star.

Since p is small it can be represent as:

`p(rad) = (1AU)/(d)` (1)

Where p(rad) is the value of in radians

However, it is better to express small angles in arcseconds

`p('') = p(rad)(180^\circ)/(\pi rad).(60')/(1^\circ).(60'')/(1')`

`p('') = 2.06x10^5 p(rad)`

`p(rad) = (p(''))/(2.06x10^5)` (2)

Then, equation 2 can be replace in equation 1:

`(p(''))/(2.06x10^5) = (1AU)/(d)`

`(d)/(1AU) = (2.06x10^5)/(p(''))` (3)

From equation 3 it can be see that
`1pc = 2.06x10^5 AU`

a) How many parsecs are there in one astronomical unit?

`1AU . (1pc)/(2.06x10^5AU)` ⇒
`4.85x10^(-6)pc`

(b) How many meters are in a parsec?

`2.06x10^(5)AU . (1.496x10^(11)m)/(1AU)` ⇒
`3.081x10^(16)m`

(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:

`x = c.t`

Where c is the speed of light (
`c = 3x10^(8)m/s`) and x is the distance that light travels in 1 year.

In 1 year they are 31536000 seconds

`x = (3x10^(8)m/s)(31536000s)`

`x = 9.46x10^(15)m`

(d) How many astronomical units in a light-year?

`9.46x10^(15)m . (1AU)/(1.496x10^(11)m)` ⇒
`63325AU`

(e) How many light-years in a parsec?

`2.06x10^(5)AU . (1ly)/(63235AU)` ⇒
`3.26ly`