(a) How many anstronomical units (AU) make 1 parsec ?
(b) Consider a sunlike star at a distance of 2 parsecs. When it is seen through a telescope with 100 magnification what should be the angular size of the star ? Sun appears to be (1/2) from the earth. Due to atmospheric fluctuations, eye cannot resolve objects smaller than 1 arc minute.
(c) Mars has approximately half of the earth's diameter. When it is closest to the earth it is at about 1/2 AU from the earth. Calculate what size it will appear when seen through the same telescope.

Explanation

By definition
1 parsec = Distance at which 1 AU long arc subtends as angle of 1"
`:.1` parsec `=((1AU)/(1..arc))`
1 deg =3600.. arc
`:.I.. arc=(pi)/(3600xx180)rad ( :. I^(@)=(pi)/(180))`
`:.` 1parsec `=(3600xx180)/(pi)AU`
`=206265AU~~2xx10^(5)AU`
(b) Sun.s diameter is `((1)/(2))` at 1AU
Therefore at 1 parsec
angular diameter of starn`=((1)/(2))/(2xx10^(5))`
With 100 magnification, it should look `15 xx 10^(-5)`. Now, due to atmospheric changes, it will still look of about l. it cannot be magnified using telescope.
(c) Given that `(D_("mars"))/(D_("earth"))=(1)/(2)`
where D represents diameter.
From answer 25(e)
We know that,
`(D_("earth"))/(D_(sun))=(1)/(100)`
`:.(D_("mars"))/(D_(sun))=(1)/(2)xx(1)/(100)` [ From eq. (1)]
At 1Au sun.s diameter `=((1)/(2))^(@)`
`:.` mar.s diameter `=(1)/(2)xx(1)/(200)=(1)/(400)`
At `(1)/(2)` AU, mar.s diameter
`=(1)/(400) xx2^(@)=((1)/(200))^(@)`
With 100 magnification, Mar.s diameter
`=(1)/(200)xx100^(@)=((1)/(2))^(@)=30.`
This is larger than resolution limit due to atmospheric fluctuations. Hence, it looks magnified.