OWAO 2023, Теория — Задача 8. Omega Sirius

• 166 pc

  Official solution: $d \approx 166$ pc; with $L \propto \mathfrak{M}^4$ the solution notes an acceptable estimate of $\sim 240$ pc.

Using spherical law of cosines we obtain the angular distance between the stars:

Both stars are main sequence stars — luminosity class V, so we can assume for such masses there is a power law relationship between mass $\mathfrak{M}$ and luminosity $L$:

Comparing the stars with the Sun ($M_{\odot} = 4.74^{\mathrm{m}}$) we obtain absolute bolometric magnitudes of them:

Absolute magnitudes in V band are

Now we can estimate distances to the stars from known absolute $M_V$ and apparent $m_V$ visual magnitudes:

The obtained values are not very accurate due to the approximations made.

Finally, the distance between stars is

Actually, since $d_2 \gg d_1$, we could conclude immediately that $d \approx d_2$.

Note. In the solution above $\omega$ CMa turned out to be noticeably fainter than in reality. This is mainly due to the fact that the mass–luminosity relation for massive stars does not provide an accurate estimate of luminosity. Due to the significant scatter in the mass-luminosity diagram for massive stars, it is acceptable to use a proportionality $L \propto \mathfrak{M}^4$, which results in a final estimate of the distance between the stars of $\sim 240$ pc.

Marking Scheme:

• Angular distance between the stars — 3 pt.
• Absolute bolometric magnitudes estimation: $L \propto \mathfrak{M}^{\alpha}$

  • $\alpha \in [3.5;\ 4.0]$ — 2 pt.
  • $\alpha \in [3.0;\ 3.5)$ — 1 pt.
• Absolute visual magnitudes — 1 pt.
• Distance to each star — 2 pt.
• Distance between the stars — 2 pt.