Question

Suppose that you now have an observation of a pulsating variable star in the Pleiades (it's like a Cephid, but lets say that it's slightly different type of variable star). Suppose that you know that this type or star obeys a period-luminosity relationship that is linear such that if the period doubles then the luminosity doubles (and if the period triples then the luminosity triples, etc). The star of this type that you observe in the Pleiades has a period of 2 days.

1. Suppose you observe a variable star of the same type in a nearby galaxy, and it also has a period of 2 days. However, to you, it appears to be only 10^-8 times as bright as the one that you saw in the Pleiades. Based on your knowledge of the inverse square law for light combined with the period-luminosity relationship I described above, what is the distance from Earth to this galaxy in parsecs?

Answer 1

According to the question, the period of a variable star in Pleiades obeys a linear relationship with luminosity. This means that if you double the period, the luminosity will also double. Hence it can be said that they are proportional to each other. This relationship can be mathematically represented as below:

where P is the Period of the star

          L is the Luminosity of the star

         is the proportionality sign

Keeping this in mind, we move forward and consider the meaning of brightness.

Luminosity is the energy coming from the star and does not depend on how far away it is from us.

Brightness is a measurement made by us as to how 'bright' is a star from a distance. Hence it is evidently dependent on luminosity and distance. The more energy a star releases(luminosity) the brighter it appears to us. Similarly if a star is far away it appears dim than if it were close by. This is just like if we look at a light bulb from a closer distance it is brighter than if we look at it from far.

The question speaks of the inverse square law of light which means the brightness of light reaching us by traveling across space depends on the square of the distance of the light source. This is because light spreads out in space. The inverse square relationship is mathematically expressed as:

where B is brightness

           d is distance from the source of light

Since Brightness also depends on Luminosity, we have to factor that as well in our mathematical expression. Since the more energy the star gives out, the more brighter it is means that they share a linear relationship just like period and luminosity. So it can now be represented as,

We can rewrite it as:

Hence,

Substitute the value of L in to get

Since the two obey a linear relationship,

where K is a constant number. This means that if P is multiplied by 2 then Bd2 also increases by a factor of 2. So eventually does not change.

Let the original star is Pleiades be give by the subscript 1 and the variable star in the nearby galaxy have the subscript 2. Then,

According to the question, B2= 10-8B1 and P1=P2=2 days

After substituting the values,

Cancelling out terms we finally get,

Taking the square root,

So the distance of the variable star in the new galaxy is 104 times the distance of earth to the Pleiades galaxy. Substitute the distance of Earth to Plaides in parasecs( experiments show it to be 133 parasec) to get the distance of Earth to a star in a nearby galaxy in parasecs( which would come to approximately 133x104 parasec)