Chapter 4, Photometry Video Solutions, An Introduction to Astronomy and Astrophysics

Problem 1

1 The unit of energy is the Joule (J) or erg.

$$
1 \mathrm{~J}=1 \mathrm{Kg} \cdot \mathrm{~m}^2 / \mathrm{s}^2, \quad 1 \mathrm{erg}=1 \mathrm{~g} \cdot \mathrm{~cm}^2 / \mathrm{s}^2
$$

Clearly, $1 \mathrm{~J}=10^7$ ergs. (a) The kinetic energy (K.E.) of a particle of mass $M$ moving at speed $v$ is equal to $M v^2 / 2$. Determine the kinetic energy of a person of mass 60 Kg moving at speed $4 \mathrm{Km} / \mathrm{hour}$. (b) Heat is also a form of energy. It is caused by the random motions of atoms and molecules of an object. As we heat an object, the kinetic energies of these particles increase. The heat capacity of water is $4,180 \mathrm{~J} / \mathrm{Kg} \cdot \mathrm{K}$ or 4,180 Joules per kilogram per degree Centigrade. This means that to increase the temperature of 1 Kg of water by $1^{\circ}$ Centigrade $(\mathrm{C})$, we need to supply 4,180 Joules of energy. Find the energy required to heat 1 Kg of water at $30^{\circ} \mathrm{C}$ (room temperature) to $100^{\circ} \mathrm{C}$.

Surjit Tewari

Numerade Educator

03:00

Problem 2

According to the special theory of relativity, a particle of mass $m$ at rest has a rest mass energy

$$
E=m c^2
$$

Determine the rest mass energy of a person of mass 60 Kg . Compare with the kinetic energy, determined in the Exercise 4.1a.

Zulfiqar Ali

Numerade Educator

03:05

Problem 3

Consider the diffuse radiation from the sky. Assume that it is uniform over the entire sky. Mathematically express the intensity received at the surface of the Earth. You can assume that roughly $20 \%$ of the total radiative energy received from the Sun, above the Earth's atmosphere, reaches the surface in the form of diffuse radiation.

Lisa Tarman

Numerade Educator

01:59

Problem 4

The luminosity of the Sun is $3.839 \times 10^{26} \mathrm{~W}$. Its apparent magnitude is $m_{\text {Sun }}=-26.81$. Find its absolute magnitude and the distance modulus.

Cindy Rodgers

Numerade Educator

05:38

Problem 5

Verify the formula for $B_\lambda$ given in Equation 4.11.

Amany Waheeb

Numerade Educator

01:51

Problem 6

Integrate the formula for blackbody specific intensity, $B_\nu$, in order to obtain an expression for the Stefan-Boltzmann constant. You can use the integral

$$
\int_0^{\infty} d x \frac{x^3}{e^x-1}=\frac{\pi^4}{15}
$$

Suzanne W.

Numerade Educator

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Problem 7

Show that the radiation flux at the surface of a star is $F=\sigma T^4$. Hence show that its total luminosity is given by Equation 4.16.

Andrew Eddins

Emory University

01:25

Problem 8

The surface temperature of the Sun is 5770 K . Write down the formula for the specific intensity, $I_\nu$, of the solar radiation received by an observer at Earth. You may assume that the Sun is an ideal blackbody. You must carefully specify both the frequency as well as angular dependence of $I_\nu$. Compute the solar flux at Earth and compare with the measured value $1,400 \mathrm{~W} / \mathrm{m}^2$.

Penny Riley

Numerade Educator

01:08

Problem 9

Find the peak wavelength and the corresponding frequency of solar radiation, given that it is a blackbody at temperature $T=5770 \mathrm{~K}$.

Narayan Hari

Numerade Educator

Problem 10

Make a rough estimate of Earth's mean temperature $T$ by assuming that it is a perfect blackbody. Assume that it absorbs all the solar radiation incident upon it and radiates it as a blackbody at temperature $T$. Assume that Earth is spherically symmetric. Numerically estimate the value of $T$.

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01:40

Problem 11

Find the peak position of wavelength and the corresponding frequency for the CMBR.

Jacob Shpiece

Numerade Educator