White Dwarf At this point our Sun starts to cool off and radiate light. A. For the surface temperature of 10? Kelvin, what is the initial luminosity of the white dwarf? B. How do you think the luminosity will change over time? C. Convert your luminosities in the below table to solar units (by dividing each by 4 × 10²? Watts). Hint: this will be very easy if you have created an Excel spreadsheet with the luminosity formula. Please upload your Excel file to the assignment folder with your lab report. D. Print out the below HR Diagram plot and label each of your luminosities in pencil from the Luminosity Table you create. (Note: if you have drawing software, you can use the image and draw the below.) a. Draw an arrow in the direction the path will follow from one phase to the next. b. Draw a path from the Super Giant phase to where you think the Sun will end up. (Remember, over time how this path would look on the HR Diagram.) Luminosity Table (to create and place in your lab report) Phase | Luminosity (in Watts) | Luminosity in Solar units ( L / 4 × 10²? Watts) 4. Main Sequence Star | | 5. Red Giant | | 6. Red Giant (before helium flash) | | 7. Red Giant (after helium flash) | | 8. Super Giant | | HR Diagram (print out, and follow instructions to fill it in, and place in your lab report)
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Step 1: To calculate the initial luminosity of the white dwarf with a surface temperature of 10^5 Kelvin, use the Stefan-Boltzmann law: L = 4πR^2σT^4. Show more…
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Luminosity of the Sun The luminosity $L$ of a star is the rate at which it radiates energy. This rate depends on the temperature $T$ and surface area $A$ of the star's photosphere (the gaseous surface that emits the light). Luminosity is modeled by the equation $L=\sigma A T^{4},$ where $\sigma$ is a constant known as the StefanBoltzmann constant, and $T$ is expressed in the absolute (Kelvin) scale for which $0 \mathrm{~K}$ is absolute zero. As with most stars, the Sun's temperature has gradually increased over the 6 billion years of its existence, causing its luminosity to slowly increase. (a) Find the rate at which the Sun's luminosity changes with respect to the temperature of its photosphere. Assume that the surface area $A$ remains constant. (b) Find the rate of change at the present time. The temperature of the photosphere is presently $5800 \mathrm{~K}\left(10,000^{\circ} \mathrm{F}\right),$ the radius of the photosphere is $r=6.96 \times 10^{8} \mathrm{~m}$ and $\sigma=5.67 \times 10^{-8} \frac{\mathrm{W}}{\mathrm{m}^{2} \mathrm{~K}^{4}}$ (c) Assuming that the rate found in (b) remains constant, how much would the luminosity change if its photosphere temperature increased by $1 \mathrm{~K}\left(1{ }^{\circ} \mathrm{C}\right.$ or $\left.1.8^{\circ} \mathrm{F}\right) ?$ Compare this change to the present luminosity of the Sun.
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The luminosity $L$ of a star is the rate at which it radiates energy. This rate depends on the temperature $T$ (in Kelvin, where $0 \mathrm{~K}$ is absolute zero) and the surface area $A$ of the star's photosphere (the gaseous surface that emits the light). Luminosity at time $t$ is given by the formula $L(t)=\sigma A T^{4}$, where $\sigma$ is a constant, known as the Stefan-Boltzmann constant. As with most stars, the Sun's temperature has gradually increased over the 5 billion years of its existence, causing its luminosity to slowly increase. For this problem, we assume that increased luminosity $L$ is due only to an increase in temperature $T .$ That is, we treat $A$ as a constant. (a) Find the rate of change of the temperature $T$ of the Sun with respect to time $t$. Write the answer in terms of the rate of change of the Sun's luminosity $L$ with respect to time $t$. (b) 4.5 billion years ago, the Sun's luminosity was only $70 \%$ of what it is now. If the rate of change of luminosity $L$ with respect to time $t$ is constant, then $\frac{\Delta L}{\Delta t}=\frac{0.3 L_{c}}{\Delta t}=\frac{0.3 L_{c}}{4.5}$, where $L_{c}$ is the current luminosity. Use differentials to approximate the current rate of change of the temperature $T$ of the Sun in degrees per century.
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(a) An astronomer obtains the spectrum of a white dwarf star and determines that it peaks in the ultraviolet at λ = 150 nm, similar to the spectra of very luminous main-sequence stars. What is the surface temperature of the white dwarf? (b) At the same time, the astronomer determines that the intrinsic luminosity of the white dwarf is only LWD = 0.01L⊙ (i.e., 1% of the luminosity of the Sun). As discussed in class, the luminosity and temperature of a star can be used to infer its radius. What is the radius of the white dwarf star? Express your answer in terms of the radius of the Sun, R⊙, and the radius of the Earth, R⊕. (i.e., how many solar [Earth] radii is this? Or, on the other hand, what fraction of the solar [Earth] radius is this?)
Adi S.
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