The brightest galaxy in the Bootes cluster of galaxies is found to have a redshift of 0.13. (Assume the speed of light, c, is 3.0 × 10 8 m s−1.)
(i)
Assuming the Hubble constant, H0, is 70 km s−1 Mpc−1 to two significant figures, calculate the distance to this galaxy. Express your answer to two significant figures and in Mpc.
The wavelength of a particular spectral line is 589 nm in the laboratory. Use your knowledge of the definition of the redshift to calculate the wavelength of this line in the spectrum of the brightest galaxy in the Bootes cluster
A hypothetical object is observed within the field of view of another galaxy cluster, which is known to be at a distance of 700 Mpc. The wavelength of the spectral feature observed in part (a) at 589 nm in the laboratory is found to be 919 nm for this object, rather than at 683 nm for the other objects in this cluster.
Without doing any calculations, explain why we can state with confidence that this object is not part of this cluster, and is actually more distant
The object is observed to have an apparent brightness of
1.2 × 10−11 W m−2. Calculate its apparent brightness if it were actually part of the cluster
(i) To calculate the distance to the galaxy with a redshift of 0.13, we can use the formula for calculating the redshift-based distance:
Distance (in Mpc) = (Redshift / Hubble constant) * c
Given the redshift as 0.13 and the Hubble constant as 70 km s^(-1) Mpc^(-1), we need to convert the Hubble constant to m s^(-1) Mpc^(-1):
Hubble constant (in m s^(-1) Mpc^(-1)) = (70 km s^(-1) Mpc^(-1)) * (1000 m/km)
Substituting the values into the formula:
Distance (in Mpc) = (0.13 / (70 * 1000)) * (3.0 * 10^8 m s^(-1))
Calculating this expression gives us the distance to the galaxy.
(ii) The redshift, z, is defined as the observed wavelength divided by the rest wavelength minus one:
z = (observed wavelength / rest wavelength) - 1
In this case, we know the rest wavelength of the spectral line is 589 nm, and we need to calculate the observed wavelength in the galaxy's spectrum. Rearranging the formula, we can solve for the observed wavelength:
observed wavelength = (z + 1) * rest wavelength
Substituting the given values of z = 0.13 and the rest wavelength = 589 nm, we can calculate the observed wavelength.
(iii) The observed wavelength of the spectral feature in the hypothetical object in the other galaxy cluster is significantly different from the wavelength observed in the Bootes cluster. This discrepancy suggests that the object is not part of the Bootes cluster but is instead more distant. Redshift is the result of the Doppler effect caused by the expansion of the universe. Objects that are farther away from us will have a greater redshift due to their greater recessional velocities. Therefore, the observed wavelength change in the hypothetical object indicates that it is farther away from the Bootes cluster.
(iv) To calculate the apparent brightness of the object if it were actually part of the cluster, we can use the inverse square law for light intensity:
Apparent Brightness = (Actual Brightness / 4π * Distance^2)
Given the apparent brightness as 1.2 x 10^(-11) W m^(-2) and the distance to the cluster as 700 Mpc, we can substitute these values into the formula and calculate the actual brightness of the object.