Stellar Magnitude & Distance

Stellar Magnitude & Distance

Astronomers measure stellar brightness on a logarithmic magnitude scale dating back to ancient Greece. The key rule: a difference of 5 magnitudes = a factor of 100 in flux.

Flux Ratio

For two stars with apparent magnitudes $m_1$ and $m_2$:

$\frac{F_1}{F_2} = 10^{(m_2 - m_1)/2.5}$

Note: smaller magnitude = brighter star (Sirius: $m = -1.46$, faintest naked-eye stars: $m \approx 6$).

Distance Modulus

The absolute magnitude $M$ is the apparent magnitude a star would have at exactly 10 parsecs. The distance modulus $\mu$ connects apparent and absolute magnitude to distance $d$ in parsecs:

$\mu = m - M = 5 \log_{10}\left(\frac{d}{10\text{ pc}}\right)$

At $d = 10$ pc: $\mu = 0$. At $d = 100$ pc: $\mu = 5$. At $d = 1000$ pc: $\mu = 10$.

Parallax

Nearby stars show an annual parallax — a tiny apparent shift as Earth orbits the Sun. The distance in parsecs is simply:

$d_{\text{pc}} = \frac{1}{p_{\text{arcsec}}}$

The Hipparcos and Gaia missions measured parallaxes for over a billion stars.

Your Task

Implement three functions. All constants must be defined inside each function.

• flux_ratio(m1, m2) — returns $F_1/F_2 = 10^{(m_2-m_1)/2.5}$
• distance_modulus(d_pc) — returns $\mu = 5 \log_{10}(d/10)$
• distance_from_modulus(mu) — returns $d = 10 \times 10^{\mu/5}$ in parsecs