Star Cluster question?
A Star cluster contains 240,000 stars. 80,000 of those stars have an apparent magnitude m=4 and the rest have an apparent magnitude m=5. The angular Diameter of the Star cluster is 1,090 arc seconds and its distance is 10,720 parsecs.
- Find the integrated apparent and absolute magnitude of the cluster.
- Find the actual diameter of the stars cluster in light years.
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Verified answer
First lets find the Apparent magnitude of the whole cluster.
L= [(n1) x (10^(1.9-0.4(m1)] + [(n2) x (10^(1.9-0.4(m2)]
m = 4.75 - 2.5 log(L)
L is the luminosity of the cluster
n1 is the number of stars with the first apparent magnitude, 80,000
m1 is the first apparent magnitude, 4
n2 is the numbers of stars with the second apparent magnitude, 160,000
m2 is the second apparent magnitude, 5
m is the integrated apparent magnitude, which is what we have to find.
L = [(80,000) x (10^(1.9-0.4(4)] + [(160,000) x (10^(1.9-0.4(5)]
L = 159,621 + 127,092.5
L = 286,713.5
m = 4.75-2.5 log(286,713.5)
m = -8.89 «------------- Apparent magnitude of the cluster
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Now lets solve for the Absolute Magnitude
M = m-5 x log(D/10)
M is the absolute magnitude
m is the apparent magnitude, -8.89
D is the distance to the cluster in parsecs, 10,720 parsecs
M = (-8.89)-5 x log(10,720/10)
M = -24.04 «---------- Absolute Magnitude of the cluster
This is a very bright cluster!
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Actual Diameter(km) = 150,000,000 x distance(parsecs) x angular diameter(arc seconds)
Actual Diameter = 150,000,000 x 10,720 x 1,090
Actual Diameter = 1.75272 × 10^15 km = 185.266 light years