Question

An astronomical unit (AU) is equal to the average distance from Earth to the Sun, about 92.9 × 10⁶ mi. A parsec (pc) is the distance at which a length of 1 AU would subtend an angle of exactly 1 second of arc. A light-year (ly) is the distance that light, traveling through a vacuum with a speed of 186 000 mi/s, would cover in 1.0 year. Express the Earth – Sun distance in (a) parsecs and (b) light- years.

Answer 1

Answer:

Explanation:

Light year is the distance , ray of light travels in one year at the speed of 186000 mi/s

Distance traveled by light in one year

= speed x time

186000 x 365 x 24 x 60 x 60 miles

= 5.8657 x 10¹² mi

the Earth – Sun distance in  light- years.

= 92.9 × 10⁶ mi. / 5.8657 x 10¹² mi  light years

= 15.84 x 10⁻⁶ light years

To calculate in terms of parsec

1 degree angle

= 60 x 60 second

π /180 radian =  60 x 60 second

1 second = 3.14 / 180 x 3600

= 4.84 x 10⁻⁶ rad

one parsec = 1AU / one second

92.9 × 10⁶ mi / one second

= 92.9 × 10⁶ mi / 4.84 x 10⁻⁶ rad

= 19.19 x 10¹² mi

the Earth – Sun distance in  parsec.

= 92.9 × 10⁶ mi. / 19.19  x 10¹² parsec.

= 4.84 x 10⁻⁶ parsec.

Answer 2

The conversion of the Earth-Sun distance from the astronomical unit (AU) to parsecs and light-years is:

a) The Earth-Sun distance in parsecs is 4.85x10⁻⁶.

b) The Earth-Sun distance in light-years is equal to 1.58x10⁻⁵.  

     

We have that:

1 AU = 92.9x10⁶ mi  

a) We know that 1 pc is the distance at which a length of 1 AU would subtend an angle of exactly 1 second of arc. We can find the distance in pc using the following trigonometric function:  

$tan(\beta) = \frac{1 AU}{pc} = \frac{92.9 \cdot 10^{6} mi}{pc}$   (1)

The angle β in radians is:  

$\beta = \frac{2 \pi \: rad}{3600*360} = 4.85 \cdot 10^{-6}$

Since the angle β is very small we can make the approximation tanβ ≅ β, so we have:                                                                              

$pc = \frac{1 AU}{4.85 \cdot 10^{-6}} = 206185.6 AU$          

So the distance in parsecs is:

$d = 1 AU = 4.85 \cdot 10^{-6} pc$

Hence, the Earth-Sun distance in parsecs is 4.85x10⁻⁶.                  

b) Knowing that a light-year (ly) is the distance that light traveling at 186000 mi/s would cover in 1 year, we can find the distance of 1 ly:

$ly = 186000 \frac{mi}{s}*\frac{365 d}{1 y}*\frac{24 h}{1 d}*\frac{3600 s}{1 h} = 5.866 \cdot 10^{12} mi$  

Now, the Earth-Sun distance (d) in ly is:

$d = \frac{1 ly}{5.866 \cdot 10^{12} mi}*92.9 \cdot 10^{6} mi = 1.58 \cdot 10^{-5} ly$    

Hence, the Earth-Sun distance is equal to 1.58x10⁻⁵ ly.    

                       

You can learn more about astronomical units here: https://brainly.com/question/989117?referrer=searchResults                              

I hope it helps you!