Solution

We use equation (2.18) to find M = m — 5 log (d/10 pc) = 5 - 5 log (100 pc/10 pc) = 5 - 5 log (10) =0

We should note that changing the distance of a star changes its apparent magnitude, but it does not change any of its colors. Because colors are defined to be differences in magnitudes, each is changed by the distance modulus. For example, using equation (2.18)

Taking the difference gives mB - mV = MB - MV

Therefore, the distance modulus never appears in the colors.

When we talk about determining an absolute magnitude, we are really only determining it over some wavelength range, corresponding to the wavelength range of the observations. We would like to have an absolute magnitude that corresponds to the total luminosity of the star. This magnitude is called the bolometric magnitude of the star. (As we will see in Chapter 4, a bolometer is a device for measuring the total energy received from an object.) For any type of star, we can define a number, called the bolometric correction (abbrevi-

Chapter summary

We saw in this chapter what can be learned from the brightness and spectrum of the continuous radiation from stars.

We introduced a logarithmic scale, the magnitude scale, for keeping track of brightness. Apparent magnitude is related to the observed energy flux from the star, and the absolute magnitude is related to the intrinsic luminosity of the star.

We saw how, even though stars are obvious to us in the visible part of the spectrum, they, and other astronomical objects, give off radiation in other parts of the spectrum. The richness of information in other parts of the spectrum is a theme that we will come back to throughout the book.

We introduced the concept of a blackbody, which is useful because the continuous spectrum of a star closely resembles that of a blackbody. Hotter bodies give off more power per unit surQuestions

2.1. Why is the magnitude scale logarithmic?

2.2. Are there any other types of measurements that we encounter in the everyday world that are logarithmic? (Hint: Think of sound.)

2.3. Why are astronomical observations potentially useful in measuring the speed of light?

2.4. What are the factors that have resulted in early astronomical observations being in the "visible" part of the spectrum?

2.5. What do we mean by "atmospheric window"?

2.6. Why was Maxwell's realization that a varying electric field can create a magnetic field important in understanding electromagnetic waves?

2.7. (a) Estimate the number of people on Earth who are exactly 2 m tall. (By "exactly" we mean to an arbitrary number of decimal places.) (b) How does this relate to the way we define the intensity function I(A)?

ated BC), which relates the bolometric magnitude to the absolute visual magnitude MV. Therefore mbol = Mv + BC (2.19)

face area than cooler ones (as described by the Stefan-Boltzmann law), and also have their spectra peaking at shorter wavelengths (as described by Wien's displacement law). We saw how attempts to understand the details of blackbody spectra (Planck's law) contributed to the idea of light coming in bundles, called photons, with specific energies. With a knowledge of blackbody spectra, we saw how stellar colors can be used to deduce stellar temperatures.

We saw how finding the distances to astronomical objects is very important, but can be quite difficult. If we don't know the distance to an object, we cannot convert its apparent brightness into a luminosity. We introduced one method of measuring distances - trigonometric parallax. It is the most direct method, but only works for nearby stars. The problem of distance determination will come up throughout the book.

2.8. What are the different ways in which the word "spectrum" is used in this chapter?

2.9. Give some examples of objects whose spectra are close to that of blackbodies.

2.10. How can we determine the temperature of a blackbody?

2.11. If the peak of a blackbody spectrum shifts to shorter wavelengths as we reach higher temperatures, how can it be that a hotter black-body gives off more energy at all wavelengths than a cooler one?

2.12. What is the evidence for the existence of photons?

2.13. Explain how we quantify the concept of color.

2.14. What is the value of using standard filters in looking at stellar spectra?

2.15. Suppose you could communicate with an astronomer on a planet orbiting a nearby star. (The astronomer is native to that planet, rather than having traveled from Earth.) You determine the distance to the star (by trigonometric parallax) to be 2 pc. The distant astronomer says that you are wrong; the distance is only 1 pc. What is the problem?

2.16. How would parallax measurement improve if we could do our observations from Mars?

2.17. As we determine the astronomical unit more accurately, how does the relationship between the AU and the parsec change?

Problems

2.1. What magnitude difference corresponds to a factor of ten change in energy flux?

2.2. One star is observed to have m = — 1 and another has m = +1. What is the ratio of energy fluxes from the two stars?

2.3. The apparent magnitude of the Sun is —26.8. How much brighter does the Sun appear than the brightest star, which has m = — 1?

2.4. (a) What is the distance modulus of the Sun? (b) What is the Sun's absolute magnitude?

2.5. Suppose two objects have energy fluxes, f and f + Af, where Af V f. Derive an approximate expression for the magnitude difference Am between these objects. Your expression should have Am proportional to Af. (Hint: Use the fact that ln (1 + x) = x when x V 1.)

2.6. Show that our definition of magnitudes has the following property: If we have three stars with energy fluxes, f1 , f2 and f3 , and we define m2 — m = 2.5 logoff m3 — m2 = 2.5 logoffs) then m3 — m = 2.5 logoff

2.7. Suppose we measure the speed of light in a laboratory, with the light traveling a path of 10 m. How accurately do you have to time the light travel time to measure c to eight significant figures?

2.8. Let A1 and A2(v^ v2) be the wavelength (frequency) limits of the visible part of the spectrum. Compare (Aa — A2)/(A1 + A2) with (v1

2.9. (a) Calculate the frequencies corresponding to the wavelengths 500.00 nm and 500.10 nm. Use these to check the accuracy of equation (2.10a). (b) Repeat the process for the second wavelength being 501.00 nm and 510.00 nm. What do you conclude?

*2.10.(a) Use equation (2.9) to derive vmax, the frequency at which I(v, T) peaks. Convert this vmax into a wavelength Amax. (b) Use equation (2.10c) to find the wavelength at which it peaks. (c) How do the results in (a) and (b) compare?

2.11. For a 300 K blackbody, over what wavelength range would you expect the Rayleigh-Jeans law to be a good approximation?

2.12. Derive an approximation for the Planck function valid for high frequencies (hv W kT).

2.13. As we will see in Chapter 21, the universe is filled with blackbody radiation at a temperature of 2.7 K. (a) At what wavelength does the spectrum of that radiation peak? (b) What part of the electromagnetic spectrum is this?

2.14. (a) We observe the blackbody spectrum from a star to peak at 400 nm. What is the temperature of the star? (b) What about one that peaks at 450 nm?

2.15. Derive an expression for the shift AA in the peak wavelength of the Planck function for a blackbody of temperature T, corresponding to a small shift in temperature, AT.

2.16. Calculate the energy per square centimeter per second reaching the Earth from the Sun.

2.17. How does the absolute magnitude of a star vary with the size of the star (assuming the temperature stays constant)?

2.18. (a) What is the energy of a photon in the middle of the visible spectrum (A = 550 nm)? (b) Approximately how many photons per second are emitted by (i) a 100W light bulb,

2.19. If we double the temperature of a blackbody, by how much must we decrease the surface area to keep the luminosity constant?

*An asterisk denotes a harder Problem or Question. The convention continues throughout the book.

2.20. (a) How does the absolute bolometric magnitude vary with the temperature of a star (assuming the radius stays constant)? (b) Does the absolute visual magnitude vary in the same way?

*2.21.For a star of radius R, whose radiation follows a blackbody spectrum at temperature T, derive an expression for the bolometric correction.

2.22. Suppose we observe the intensity of a black-body, I0, in a narrow frequency range centered at v0. Find an expression for T, the temperature of the blackbody in terms of I0 and v0. (a) First do it in the Rayleigh-Jeans limit and (b) in the general case.

*2.23.Suppose we receive light from a star for which the received energy flux is given by the function f(A). Suppose we observe the star through a filter for which the fraction of light transmitted is t (A). Derive an expression for the total energy detected from the star. (Hint: Start by thinking of the energy detected in a small wavelength range.)

2.24. What is the distance to a star whose parallax is 0.1 arc sec?

2.25. Derive an expression for the distance of an object as a function of the parallax angle seen by your eyes?

2.26. (a) If we can measure parallaxes as small as 0.1 arc sec, what is the greatest distance that can be measured using the method of trigonometric parallaxes? (b) By what factor will the volume of space over which we can measure parallax change if we can measure to 0.001 arc sec? (c) Why is the volume of space important?

2.27. If we lived on Mars instead of the Earth, how large would the parsec be?

2.28. Suppose we discover a planet orbiting a nearby star. The distance to the star is 3 pc. We observe the angular radius of the planet's orbit to be 0.1 arc sec. How many AU from the star is the planet? (Hint: You can solve this problem by "brute force", converting all the units. For an easier solution, think about what the answer would be if the star were 1 pc from us and the angular radius of the orbit were 1 arc sec, and then scale the result accordingly.)

2.29. Derive an expression for the distance to a star in terms of its distance modulus.

2.30. If we make a 0.05 magnitude error in measuring the apparent magnitude of a star, what error does that introduce in our distance determination (assuming its absolute magnitude is known exactly)?

Computer problems

2.1. Make a fourth column for Table 2.1, showing the range of photon frequencies for each part of the spectrum. Make a fifth column showing the range of photon energies for each part of the spectrum. Make a sixth column showing the temperatures that blackbodies would have to peak at the wavelengths corresponding to the boundaries between the parts of the spectrum

2.2. Make a graph of the magnitude difference MB — MV as a function of temperature for a temperature range of 3000 K to 30 000 K. To simplify the calcu lation you may assume that magnitudes are determined in a narrow range of wavelengths around the peak of each filter.

2.3. For the Sun, plot the difference between the Rayleigh-Jeans approximation and the Planck formula, as a function of wavelength, for wavelengths in the visible part of the spectrum.

2.4. For the Sun, calculate the energy given off over the wavelength bands that correspond to the U, B and V filters. Use this to estimate the colors U — B and B — V.

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