The peak luminosity of a white dwarf supernova is around 10^10LSun and it remains above 10^8LSun for about 150 days. In comparison, the luminosity of a bright Cepheid variable star is about 10000LSun.The Hubble Space Telescope is sensitive enough to make accurate measurements of apparent brightness for Cepheid variables at distances up to about 100 million light-years.
Estimate the distance of a fading white dwarf supernova of luminosity 10^8LSun whose apparent brightness is comparable to that of a bright Cepheid variable star 100 million light-years from Earth.
To estimate the distance of a fading white dwarf supernova, we can use the concept of luminosity-distance relationship. The relationship between luminosity (L) and distance (d) is given by the inverse square law:
L = 4πd^2 B,
where B is the apparent brightness.
We are given that the peak luminosity of the white dwarf supernova is 10^10 Lsun (solar luminosities) and it remains above 10^8 Lsun for about 150 days. We are also told that the luminosity of a bright Cepheid variable star is about 10000 Lsun. The Hubble Space Telescope can accurately measure the apparent brightness of Cepheid variables up to a distance of 100 million light-years.
Using the given information, let's calculate the apparent brightness (B) of the white dwarf supernova at 10^8 Lsun, which is comparable to the bright Cepheid variable star:
B_supernova = B_Cepheid = 10000 Lsun.
Now, let's use the luminosity-distance relationship to calculate the distance (d) of the fading white dwarf supernova:
10^8 Lsun = 4πd^2 B_supernova.
Rewriting this equation for d:
d^2 = (10^8 Lsun) / (4πB_supernova).
d^2 = (10^8 Lsun) / (4π * 10000 Lsun).
d^2 = 10^8 / (4π * 10000).
d^2 = 7957.75.
Taking the square root of both sides, we get:
d ≈ √7957.75.
d ≈ 89.21 light-years.
Therefore, the estimated distance of the fading white dwarf supernova, with a luminosity of 10^8 Lsun and comparable apparent brightness to a bright Cepheid variable star, is approximately 89.21 light-years.
To estimate the distance of a fading white dwarf supernova with a luminosity of 10^8LSun, we need to compare its apparent brightness to that of a bright Cepheid variable star at a known distance.
Given that the Hubble Space Telescope can make accurate measurements of the apparent brightness of Cepheid variables at distances up to about 100 million light-years, we can assume that the apparent brightness of a Cepheid variable star at this distance is detectable.
First, we need to determine the ratio of the luminosities between the white dwarf supernova (10^8LSun) and the bright Cepheid variable star (10000LSun).
Luminosity Ratio = (White Dwarf Supernova Luminosity) / (Cepheid Variable Star Luminosity)
= (10^8LSun) / (10000LSun)
= 10^4
Given that the apparent brightness of the bright Cepheid variable star at a distance of 100 million light-years is detectable, we can assume that the apparent brightness of the fading white dwarf supernova at an unknown distance is comparable to that of the Cepheid variable star.
Using the inverse square law of light, which states that the apparent brightness is inversely proportional to the square of the distance, we can set up the following equation:
(Apparent Brightness of Fading White Dwarf Supernova) / (Apparent Brightness of Cepheid Variable Star) = (Distance of Cepheid Variable Star)^2 / (Distance of Fading White Dwarf Supernova)^2
Let's substitute the known values into the equation:
(10^8LSun) / (10000LSun) = (100,000,000 light-years)^2 / (Distance of Fading White Dwarf Supernova)^2
We can solve for the Distance of Fading White Dwarf Supernova:
(Distance of Fading White Dwarf Supernova)^2 = (100,000,000 light-years)^2 / ((10^8LSun) / (10000LSun))
Taking the square root of both sides of the equation:
Distance of Fading White Dwarf Supernova = Square Root((100,000,000 light-years)^2 / ((10^8LSun) / (10000LSun)))
Using a calculator or computational tool, we can calculate the approximate distance of the fading white dwarf supernova.