Distance to a star using parallax. Estimate the distance D to a star if the angle $\theta$ in Fig.44-11 is measured to be $89.99994^{\circ}$ .

APPROACH From trigonometry, $\tan \phi=d / D$ in Fig. 44-11. The Sun-Earth distance is $d=1.5 \times 10^{8} \mathrm{~km}$ .

Question

Distance to a star using parallax. Estimate the distance D to a star if the angle [latex] heta[/latex] in Fig.44-11 is measured to be [latex]89.99994^{\circ}[/latex].

APPROACH From trigonometry, [latex] an \phi=d / D[/latex] in Fig. 44-11. The Sun-Earth distance is [latex]d=1.5 imes 10^{8} \mathrm{~km}[/latex].

Accepted Answer

The angle [latex]\phi=90^{\circ}-89.99994^{\circ}=0.00006^{\circ}[/latex], or about [latex]1.0 imes 10^{-6}[/latex] radians. We can use [latex] an \phi \approx \phi[/latex] since [latex]\phi[/latex] is very small. We solve for D in [latex] an \phi=d / D[/latex]. The distance D to the star is

[latex]D=\frac{d}{ an \phi} \approx \frac{d}{\phi}=\frac{1.5 imes 10^{8} \mathrm{~km}}{1.0 imes 10^{-6} \,\mathrm{rad}}=1.5 imes 10^{14} \mathrm{~km},[/latex]

or about 15 ly.

Question 44.6: Distance to a star using parallax. Estimate the distance D t......

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