PX158 - PS2

Q1

A star at a distance of $d = 135 p c$ has an apparent magnitude, $m = 17.5$ . What is its absolute magnitude?

\begin{aligned} m - M & = 5 \log_{10} (\frac{d}{10 p c}) \\ M & = 11.8 \end{aligned}

The faintest objects ever detected at optical wavelengths have apparent magnitudes, $m = 31$ . Calculate the flux from an object of this magnitude, and convert your result into a photon flux (i.e. the number of photons per unit time per unit area at Earth).
let object 1 be the faint star and object 2 be the sun:

\begin{aligned} m_{1} - m_{2} & = - 2.5 \log_{10} (\frac{f_{1}}{f_{2}}) \\ \frac{f_{1}}{f_{2}} & = 10^{- \frac{m_{1} - m_{2}}{2.5}} \\ = 10^{- \frac{31 - (- 26.8)}{2.5}} \\ f_{1} & = 1350 \times 10^{- \frac{31 - (- 26.8)}{2.5}} \\ = 10^{- 20} W m^{- 2} \\ ∴ f_{1} & = 0.025 p h o t o n s s^{- 1} m^{- 2} \end{aligned}

On $J a n 23, 1999$ a remarkable gamma-ray burst (GRB) briefly peaked at $m = 9$ despite being some $3 G p c = 3 \times 10^{9} p c$ from Earth. Calculate its luminosity as a multiple of the Sun’s luminosity.

\begin{aligned} m - M & = 5 \log_{10} (\frac{d}{10 p c}) \\ M & = - 33.4 \end{aligned}

\begin{aligned} m_{1} - m_{2} & = - 2.5 \log_{10} (\frac{f_{1}}{f_{2}}) \\ M_{1} - M_{2} & = - 2.5 \log_{10} (\frac{L_{1}}{L_{2}}) \\ ∴ L_{1} & = 2 \times 10^{15} L_{⊙} \end{aligned}

A lens of diameter, $D = 100 m m$ and focal length, $f = 200 m m$ is used to focus an image of the Sun onto a piece of paper. What is the size of the Sun’s image and what is the incident power per unit area within the image?

l = f α = 200 \times 30 \times \frac{π}{60 \times 180} m m = 1.8 m m

\frac{P}{A} = 1350 (\frac{π {(\frac{D}{2})}^{2}}{π {(\frac{l}{2})}^{2}}) = 4 \times 10^{6} W m^{- 2}

A star changes brightness by $+ 1$ magnitude. What is the corresponding change in its flux?

m_{1} - m_{2} = - 2.5 \log_{10} (\frac{f_{1}}{f_{2}})

A star’s flux increase by a factor $2$ . What is the corresponding change in its magnitude?

Δ m = - 0.75

Calculate the total absolute magnitude of a star cluster containing of $500000$ Sun-like stars.
$M_{⊙} = + 4.8$

M = - 9.45

Show that an increase of $1 %$ in brightness corresponds to a magnitude change $∆ m \approx - 0.01$

m_{1} - m_{2} = - 2.5 \log_{10} (\frac{f_{1}}{f_{2}})

A star brightens by a factor of 2.3 in flux measured through a B filter, and at the same time by 1.5 in flux measured through a V filter. Calculate the change in its $B - V$ colour.

Δ B = - 2.5 \log_{10} (2.3) = - 0.904

Δ V = - 2.5 \log_{10} (1.5) = - 0.44

∴ B - V = Δ B - Δ V = - 0.464

The daytime sky brightness on a cloudless day when the Sun is well above the horizon is roughly equivalent to a star of magnitude $3.5 p e r s q u a r e a r c s e c o n d$ . Use this to estimate the limiting magnitude of stars and planets for them to be visible in the daytime ( $(i)$ by unaided eye, $(i i)$ by eye with the aid of a telescope. (Assume that the target flux has to exceed $10 %$ of the background flux, with the patch of background a square equal on each side to the minimum resolvable angle in each case.)