Distance & Parallax

Useful constants: $1\ \mathrm{pc} = 3.09 \times 10^{18}\ \mathrm{cm} = 3.26\ \mathrm{ly}$ ; $1\ \mathrm{rad} = 206{,}265''$ ; $1\ \mathrm{AU} = 1.5 \times 10^{13}\ \mathrm{cm}$ ; $L_\odot = 3.8 \times 10^{33}\ \mathrm{erg\,s^{-1}}$ .

Conceptual

1. Why is distance measurement hard? Explain why stellar parallax angles are so tiny. Use the small-angle relation to estimate the parallax of a star at 30 pc.

(a) Show that $p \approx 1/d$ (pc, arcsec), so a star at 30 pc has $p \approx 0.03''$ .
(b) Convert $0.03''$ to microarcseconds and compare to Gaia’s precision ( $\sim 10\ \mu$ as).
(c) Explain why measuring such tiny angles requires space telescopes.

2. Inverse-square law and stellar brightness. Two identical stars have the same luminosity $L$ ; Star A is twice as far as Star B.

(a) By what factor is Star A fainter (in observed flux)?
(b) If Star B has flux $F_B$ , what is Star A’s flux?
(c) Explain why the inverse-square law is universal — it applies to any radiation spreading isotropically from a point source.

Calculation

3. Parallax-distance conversion. Convert each parallax to distance in parsecs, then light-years:

(a) Sirius: $p = 0.379''$
(b) Vega: $p = 0.129''$
(c) Altair: $p = 0.198''$

4. Computing stellar luminosity. A star has $p = 0.050''$ and flux $F = 5 \times 10^{-13}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ .

(a) Calculate its distance in parsecs and centimeters.
(b) Use $L = 4\pi d^2 F$ to compute the luminosity in erg/s.
(c) Express it as a fraction of $L_\odot$ .
(d) Sanity check: is this reasonable for a typical star?

5. Error propagation from parallax to luminosity. Gaia measures $p = 0.100 \pm 0.001''$ (1% uncertainty).

(a) Find the distance and its fractional uncertainty.
(b) Photometry gives $F = 1.0 \times 10^{-11} \pm 0.05 \times 10^{-11}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ (5%).
(c) Compute $L = 4\pi d^2 F$ . What fractional uncertainty in $L$ comes from parallax alone?
(d) Combined with the 5% photometry uncertainty, what is the total fractional uncertainty in $L$ ?
(e) Why is precise parallax so critical for accurate luminosity?

Synthesis

6. The cosmic distance ladder — parallax vs. standard candles.

(a) Hipparcos (1 mas precision): maximum reliable distance for parallax of (i) $1''$ ? (ii) $0.001''$ ?
(b) Gaia DR2 (10 μas): redo (a).
(c) Gaia future (1 μas): redo (a).
(d) A Cepheid in Andromeda ( $\sim 770$ $\sim 770$ kpc) has $F = 10^{-14}\ \mathrm{erg\,cm^{-2}\,s^{-1}}$ $F = 1 0^{- 14} erg c m^{- 2} s^{- 1}$ and known $L = 10{,}000\,L_\odot \approx 3.8 \times 10^{37}\ \mathrm{erg/s}$ $L = 10, 000 L_{⊙} \approx 3.8 \times 1 0^{37} erg/s$ .
- (i) Verify whether flux and luminosity are consistent at 770 kpc.
- (ii) Why can’t we measure Andromeda’s distance with parallax, even with Gaia?
- (iii) Explain how the distance ladder combines parallax (nearby) with standard candles (distant).

7. From parallax and photometry to the HR diagram. Three stars (smaller magnitude = brighter):

Star	Parallax (arcsec)	Apparent magnitude
A	0.10	5.0
B	0.010	5.0
C	0.010	3.0

(a) Rank the stars by distance (nearest to farthest).
(b) Rank by intrinsic luminosity (brightest to dimmest), and explain.
(c) Why is it impossible to determine intrinsic luminosity from apparent brightness alone? What’s missing?
(d) Write out the complete Observable → Model → Inference sequence for placing a star on the HR diagram using parallax and photometry.