Angle measurement of Polaris

Galileo Gallei measured the angular positions of the stars with his telescope. To adjust his telescope, he initially aimed at the Pole Star, which is always at the same position in the sky at an angle of α\alpha with the horizontal.

On a cold winter's night of 3 °C,\SI{-3}{\celsius}, he conducted his measurements inside his study room at 20 °C.\SI{20}{\celsius}. The starlight fell through his window and refracted twice on the glass. From his room, Polaris appeared at an angle of β\beta with the horizon.

What is the angle difference δ=βα\delta = \beta - \alpha between measurements inside and outside?

Assumptions:

  • Galileo's study room was in Venice, at a latitude of ϕ45 °.\phi \approx \SI{45}{\degree}.
  • Air is an ideal gas with the specific gas constant Rair300 J/(kgK).R_\text{air}^\ast \approx 300 \text{ J}/(\text{kg} \cdot \text{K}).
  • Air is a dielectric with an electric susceptibility of χair=εair1=cρair(where c103 m3/kg). \chi_\text{air} = \varepsilon_\text{air} - 1 = c \cdot \rho_\text{air}\, \big(\text{where } c \approx 10^{-3} \text{ m}^3/\text{kg}\big).
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