Diagram 15-1
-Refer to Diagram 15-1.A type I Cepheid has been located in an open cluster.The period of the Cepheid variable is 30 days,and the variable's apparent visual magnitude is 10.What is the distance to this open cluster?

Question

Diagram 15-1 
-Refer to Diagram 15-1.A type I Cepheid has been located in an open cluster.The period of the Cepheid variable is 30 days,and the variable's apparent visual magnitude is 10.What is the distance to this open cluster?

Quizplus · Accepted Answer

The period-luminosity relation for Cepheid variables is given by:$$\log L = 1.96\log P - 0.20$$where L is the luminosity in solar luminosities and P is the period in days.The apparent visual magnitude of the Cepheid variable is 10, which means that its absolute visual magnitude is:$$M_v = m_v - 5\log d + 5$$where d is the distance to the Cepheid variable in parsecs.Substituting the values of L, P, and M_v into the above equations, we get:$$\log L = 1.96\log 30 - 0.20 = 4.28$$$$M_v = 10 - 5\log d + 5 = -5\log d + 5$$Setting the two equations equal to each other, we get:$$-5\log d + 5 = 4.28$$Solving for d, we get:$$d = 10^{0.28} = 1.90 	ext{ kpc} = 10,000 	ext{ pc}$$Therefore, the distance to the open cluster is 10,000 pc.

Diagram 15-1 -Refer to Diagram 15-1.A Type I Cepheid Has Been Located