Answer:
0.61
Explanation:
To calculate Vega's absolute magnitude [tex]\bold{( M )}[/tex], we use the distance modulus formula, which relates apparent magnitude [tex]\bold{( m )}[/tex], absolute magnitude [tex]\bold{( M )}[/tex], and distance in parsecs [tex]\bold{( d )}[/tex]:
[tex] \large\boxed{\boxed{\sf m - M = 5 \log_{10}(d) - 5 }}[/tex]
First, we need to convert the distance from light years to parsecs.
Given that 1 parsec (pc) is approximately 3.262 light years (ly), we calculate:
[tex] \sf d = \dfrac{25.05 \, \textsf{ly}}{3.262 \, \textsf{ly/pc}} \approx 7.68 \, \textsf{pc} [/tex]
Nextsf, we use the distance modulus formula:
[tex] \sf m - M = 5 \log_{10}(d) - 5 [/tex]
Substitute [tex] \bold{ m = 0.0305 }[/tex] and [tex] \bold{ d = 7.68 }[/tex]:
[tex] \sf 0.0305 - M = 5 \log_{10}(7.68) - 5 [/tex]
[tex] \sf 0.0305 - M = 5 \times 0.88536 - 5 [/tex]
[tex] \sf 0.0305 - M = 4.4268 - 5 [/tex]
[tex] \sf 0.0305 - M = -0.5732 [/tex]
Solving for [tex] \bold{ M }[/tex]:
[tex] \sf M = 0.0305 + 0.5732 [/tex]
[tex] \sf M = 0.6037 [/tex]
[tex] \sf M = 0.61 \textsf{(in 2 d.p.)} [/tex]
Thus, the absolute magnitude of Vega is approximately:
[tex] \sf M \approx 0.61 [/tex]