PX158 - PS1

Q1

The bright star Sirius has a declination of $- 16.75 °$ . What is the highest altitude that an observer in Rome $(l a t i t u d e = + 42 °$ ) would be able to observe Sirius? Does the time of year affect your answer?
here, $δ = - 16.75 °, l a t = + 42 °$

\begin{aligned} a l t_{m a x} & = 90 ° - (l a t - δ) \\ = 31.25 ° \end{aligned}

How many degrees does the Sun move each day relative to the fixed stars on the celestial sphere?

\frac{360 °}{365.25} \approx 0.985 ° / d a y

The Sun and nearby stars orbit the centre of our Galaxy once every 230 million years. Calculate how rapidly a coordinate frame locked to nearby stars rotates relative to the non-rotating frame defined by distant galaxies. Express your answer in arcseconds per year ( $^{''} y r^{- 1}$ ).

\frac{360 \times 3600}{230 \times 10^{6}} = {0.0056}^{^{″}} y r^{- 1}

Stars exhibit random motions with respect to the Sun which cause their positions to change. This is known as “proper motion”. Calculate how rapidly a star $30 p c$ away and moving at $50 k m s^{- 1}$ perpendicular to the line-of sight changes position. Again express your answer in $^{''} y r^{- 1}$ .
here, $l = 50 \times 10^{3} \times (365 \times 24 \times 3600), d = 30 \times 3.086 \times 10^{16}$

α \approx \frac{l}{d} \times (\frac{180}{π} \times 3600) \approx {0.35}^{^{″}} y r^{- 1}

The European Space Agency’s Gaia mission is designed to measure the positions of stars to a precision of $2 \times 10^{- 5''}$ . This is small enough to detect extra-solar planets through the position wobble of their host stars. Given that, owing to the gravitational pull of Jupiter, the Sun executes an orbit of radius, $750000 k m$ , estimate the maximum distance in parsecs to which Gaia could detect that a Sun-like star hosts a Jupiter-like planet.
here, $α = 3 \times 10^{- 5^{″}}, l = 2 r = 1.5 \times 10^{6} k m = 0.01 A U$

d = \frac{0.01 A U}{2 \times 10^{- 5^{″}}} = 500 p c

From the ground, parallaxes are measured relative to background stars close to the target on the sky. If these are not infinitely distant, then the parallax to the foreground object will be underestimated and its distance will be overestimated. Calculate the distance that will be measured to a star at a true distance of $40 p c$ if the background stars are at a distance of $500 p c$ and this effect is not allowed for.
here, two stars at distances, $d_{1} = 40 p c$ and $d_{2} = 500 p c$ , making parallax angles, $α_{1}$ and $α_{2}$

α_{1} - α_{2} = \frac{1}{d_{1}} - \frac{1}{d_{2}} = {0.023}^{^{″}}

d_{1}^{'} = \frac{1}{{0.023}^{^{″}}} = 43.5 p c

The planet Mars has a radius of $3390 k m$ and orbits $1.523 A U$ from the Sun. Assuming that Earth and Mars have circular, co-planar orbits, calculate the maximum and minimum angle subtended by Mars from Earth. Why would the minimum angle be particularly hard to measure?
here, $d_{E M, m i n} = (1.523 - 1) A U$ , and $d_{E M, m a x} = (1.523 + 1) A U$

α_{m a x} = \frac{2 \times 3390}{0.523 \times 1.5 \times 10^{11}} \times (\frac{180}{π} \times 3600) \approx {17.87}^{^{″}}

α_{m i n} = \frac{2 \times 3390}{2.523 \times 1.5 \times 10^{11}} \times (\frac{180}{π} \times 3600) \approx {3.71}^{^{″}}

$α_{m i n}$ happens when mars in on the other side of the sun, thus making it hard to measure

Two observers at diametrically opposite points on Earth observe the Moon simultaneously. What difference in angle of the Moon relative to the fixed stars will they measure? (Radius of Earth RE = $6371 k m$ ; Earth–Moon distance $384000 k m$ )

α = 0.0334 \times \frac{180}{π} \approx 1.9 °