The Light the Earth Gets From the Sun is a Continuous Spectrum
Problem 1
In $1672,$ an international effort was made to measure the parallax angle of Mars at the time of opposition, when it was closest to Earth; see the below figure.
(a) Consider two observers who are separated by a baseline equal to Earth's diameter. If the difference in their measurements of Mars's angular position is $33.6 "$, what is the distance between Earth and Mars at the time of opposition? Express your answer both in units of $\mathrm{m}$ and in AU.
(b) If the distance to Mars is to be measured to within $10 \%$, how closely must the clocks used by the two observers be synchronized? Hint: Ignore the rotation of Earth. The average orbital velocities of Earth and Mars are $29.79 \mathrm{km} \mathrm{s}^{-1}$ and $24.13 \mathrm{km} \mathrm{s}^{-1},$ respectively.
John J.
Numerade Educator
Problem 2
At what distance from a 100 -W light bulb is the radiant flux equal to the solar irradiance?
Umar Sohail Q.
Numerade Educator
Problem 3
The parallax angle for Sirius is $0.379^{\prime \prime}$
(a) Find the distance to Sirius in units of (i) parsecs; (ii) light-years; (iii) AU; (iv) m.
(b) Determine the distance modulus for Sirius.
Umar Sohail Q.
Numerade Educator
Problem 4
Using the information in Example 6.1 and Problem $3,$ determine the absolute bolometric magnitude of Sirius and compare it with that of the Sun. What is the ratio of Sirius's luminosity to that of the Sun?
Umar Sohail Q.
Numerade Educator
Problem 5
(a) The Hipparcos Space Astrometry Mission was able to measure parallax angles down to nearly $0.001^{\prime \prime} .$ To get a sense of that level of resolution, how far from a dime would you need to be to observe it subtending an angle of $0.001^{\prime \prime} ?$ (The diameter of a dime is approximately $1.9 \mathrm{cm} .$
(b) Assume that grass grows at the rate of $5 \mathrm{cm}$ per week.
i. How much does grass grow in one second?
ii. How far from the grass would you need to be to see it grow at an angular rate of $0.000004^{\prime \prime}(4$ microarcseconds ) per second? Four microarcseconds is the estimated angular resolution of SIM, NASA's planned astrometric mission.
Linda W.
Numerade Educator
Problem 6
Derive the relation
\[m=M_{\mathrm{Sun}}-2.5 \log _{10}\left(\frac{F}{F_{10, \odot}}\right)\]
Linda W.
Numerade Educator
Problem 7
A $1.2 \times 10^{4} \mathrm{kg}$ spacecraft is launched from Earth and is to be accelerated radially away from the Sun using a circular solar sail. The initial acceleration of the spacecraft is to be 1 g. Assuming a flat sail, determine the radius of the sail if it is
(a) black, so it absorbs the Sun's light.
(b) shiny, so it reflects the Sun's light.
Hint: The spacecraft, like Earth, is orbiting the Sun. Should you include the Sun's gravity in your calculation?
Linda W.
Numerade Educator
Problem 8
The average person has $1.4 \mathrm{m}^{2}$ of skin at a skin temperature of roughly $306 \mathrm{K}\left(92^{\circ} \mathrm{F}\right)$. Consider the average person to be an ideal radiator standing in a room at a temperature of $293 \mathrm{K}\left(68^{\circ} \mathrm{F}\right)$
(a) Calculate the energy per second radiated by the average person in the form of blackbody radiation. Express your answer in watts
(b) Determine the peak wavelength $\lambda_{\ max }$ of the blackbody radiation emitted by the average person. In what region of the electromagnetic spectrum is this wavelength found?
(c) A blackbody also absorbs energy from its environment, in this case from the 293 -K room. The equation describing the absorption is the same as the equation describing the emission of blackbody radiation, Eq. ( 16 ). Calculate the energy per second absorbed by the average person, expressed in watts. $$L=A \sigma T^{4}$$ (d) Calculate the net energy per second lost by the average person via blackbody radiation.
Umar Sohail Q.
Numerade Educator
Problem 9
Consider a model of the star Dschubba $(\delta \mathrm{Sco}),$ the center star in the head of the constellation Scorpius. Assume that Dschubba is a spherical blackbody with a surface temperature of $28,000 \mathrm{K}$ and a radius of $5.16 \times 10^{9} \mathrm{m}$. Let this model star be located at a distance of $123 \mathrm{pc}$ from Earth. Determine the following for the star:
(a) Luminosity.
(b) Absolute bolometric magnitude.
(c) Apparent bolometric magnitude.
(d) Distance modulus.
(e) Radiant flux at the star's surface.
(f) Radiant flux at Earth's surface (compare this with the solar irradiance).
(g) Peak wavelength $\lambda_{\max }$
Linda W.
Numerade Educator
Problem 10
(a) Show that the Rayleigh-Jeans law (Eq. 20 ) is an approximation of the Planck function $B_{\lambda}$ in the limit of $\lambda \gg h c / k T .$ (The first-order expansion $e^{x} \approx 1+x$ for $x \ll 1$ will be useful.) Notice that Planck's constant is not present in your answer. The Rayleigh-Jeans law is a classical result, so the "ultraviolet catastrophe" at short wavelengths, produced by the $\lambda^{4}$ in the denominator, cannot be avoided. The Continuous Spectrum of Light: Problem Set $$B_{\lambda}(T) \simeq \frac{2 c k T}{\lambda^{4}}, \quad(\text { valid only if } \lambda \text { is long })$$
(b) Plot the Planck function $B_{\lambda}$ and the Rayleigh-Jeans law for the $\operatorname{Sun}\left(T_{\odot}=5777 \mathrm{K}\right)$ on the same graph. At roughly what wavelength is the Rayleigh-Jeans value twice as large as the Planck function?
Linda W.
Numerade Educator
Problem 11
Show that Wien's expression for blackbody radiation (Eq. 21 ) follows directly from Planck's function at short wavelengths. $$\left.B_{\lambda}(T) \simeq a \lambda^{-5} e^{-b / \lambda T}, \quad \text { (valid only if } \lambda \text { is short }\right)$$
Linda W.
Numerade Educator
Problem 12
Derive Wien's displacement law, Eq. ( 15 ), by setting $d B_{\lambda} / d \lambda=0 .$ Hint: You will encounter an equation that must be solved numerically, not algebraically. $$\lambda_{\max } T=0.002897755 \mathrm{m} \mathrm{K}$$
Linda W.
Numerade Educator
Problem 13
(a) Use Eq. ( 24 ) to find an expression for the frequency $v_{\max }$ at which the Planck function $B_{v}$ attains its maximum value. (Warning: $\left.v_{\max } \neq c / \lambda_{\max } .\right)$ $$B_{v}(T)=\frac{2 h v^{3} / c^{2}}{e^{h v / k T}-1}$$ (b) What is the value of $v_{\max }$ for the Sun?
(c) Find the wavelength of a light wave having frequency $v_{\max } .$ In what region of the electromagnetic spectrum is this wavelength found?
Linda W.
Numerade Educator
Problem 14
(a) Integrate Eq. ( 27 ) over all wavelengths to obtain an expression for the total luminosity of a blackbody model star. Hint: $$\int_{0}^{\infty} \frac{u^{3} d u}{e^{u}-1}=\frac{\pi^{4}}{15}$$ $$=\frac{8 \pi^{2} R^{2} h c^{2} / \lambda^{5}}{e^{h c / \lambda k T}-1} d \lambda$$ (b) Compare your result with the Stefan-Boltzmann equation $(17),$ and show that the StefanBoltzmann constant $\sigma$ is given by $$\sigma=\frac{2 \pi^{5} k^{4}}{15 c^{2} h^{3}}$$ $$L=4 \pi R^{2} \sigma T_{e}^{4}$$ (c) Calculate the value of $\sigma$ from this expression.
Umar Sohail Q.
Numerade Educator
Problem 15
Use the data in Appendix: Stellar Data, to answer the following questions.
(a) Calculate the absolute and apparent visual magnitudes, $M_{V}$ and $V$, for the Sun.
(b) Determine the magnitudes $M_{B}, B, M_{U},$ and $U$ for the Sun.
(c) Locate the Sun and Sirius on the color-color diagram in Fig. $11 .$ Refer to Example 6.1 for the data on Sirius.
Linda W.
Numerade Educator
Problem 16
Use the filter bandwidths for the $U B V$ system in section 6 of "The Continuous Spectrum of Light" and the effective temperature of $9600 \mathrm{K}$ for Vega to determine through which filter Vega would appear brightest to a photometer [i.e., ignore the constant $C \text { in Eq. }(31)]$. Assume that $\mathcal{S}(\lambda)=1$ inside the filter bandwidth and that $\mathcal{S}(\lambda)=0$ outside the filter bandwidth. $$U=-2.5 \log _{10}\left(\int_{0}^{\infty} F_{\lambda} \mathcal{S}_{U} d \lambda\right)+C_{U}$$
Problem 17
Evaluate the constant $C_{\text {bol }}$ in Eq. (3.32) by using $m_{\mathrm{Sun}}=-26.83$
Linda W.
Numerade Educator
Problem 18
Use the values of the constants $C_{U-B}$ and $C_{B-V}$ found in Example 6.2 of "The Continuous Spectrum of Light" to estimate the color indices $U-B$ and $B-V$ for the Sun.
Problem 19
Shaula ( $\lambda$ Scorpii) is a bright $(V=1.62)$ blue-white subgiant star located at the tip of the scorpion's tail. Its surface temperature is about $22,000 \mathrm{K}$
(a) Use the values of the constants $C_{U-B}$ and $C_{B-V}$ found in Example 6.2 of "The continuous spectrum of Light" to estimate the color indices $U-B$ and $B-V$ for Shaula. Compare your answers with the measured values of $U-B=-0.90$ and $B-V=-0.23$
(b) The Hipparcos Space Astrometry Mission measured the parallax angle for Shaula to be $0.00464^{\prime \prime} .$ Determine the absolute visual magnitude of the star. (Shaula is a pulsating star, belonging to the class of Beta Cephei variables. its magnitude varies between $V=1.59$ and $V=1.65$ with a period of 5 hours 8 minutes, its color indices also change slightly.)
Source: https://www.numerade.com/books/chapter/the-continuous-spectrum-of-light/
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