6. "MHz" is the abbreviation for a megahertz, which is one million hertz (106 Hz). First convert 893.07 MHz into hertz:
Now find the wavelength that corresponds to this frequency. (Remember that Hz can also be written as s-1.)
(So these radio waves have wavelengths about a foot long.)
7. The Stefan-Boltzmann law says that the amount of energy (in the form of light) emitted each second by one square-meter of a blackbody's surface is proportional to the fourth power of the temperature:
So as the temperature increases, the amount of light emitted by each square meter of the object's surface (the flux) will increase. Doubling the Kelvin temperature will cause the flux to increase by a factor of 24 = 16.
Wien's law says that the wavelength emitted most strongly by a blackbody (lmax) is inversely proportional to the temperature:
So as the temperature increases, lmax will decrease proportionally. For visible light, longer wavelenths are redder and shorter wavelengths are bluer, so this explains why the overall color of a relatively cool object is reddish, while that of a hotter object is bluish.
8. A blackbody is a hypothetical object that absorbs all light that you shine on to its surface, and reflects none of it. Since it reflects no light, we say that it's "black." However, it is important to realize that objects can give off light in two ways: by reflecting light that is shining on to it from an outside source, and by generating its own light ("glowing") due to its temperature. Though a blackbody will reflect no light from an outside source, it will glow with a continuous spectrum due to its temperature, as described by Wien's law and the Stefan-Boltzmann law. Thus, a blackbody may or may not look black, depending on its temperature.
A blackbody at room temperature (around 300 K) glows very little (Stefan-Boltzmann law), and most of the light by which it glows is in the infrared (Wien's law -- lmax is in the infrared). So it will look black to your eyes. If you shine a flashlight on it, none of that light will be reflected to your eyes, so the object will still look black.
A blackbody at 6,000 K (the approximate surface temperature of the Sun) glows much more at all wavelengths (Stefan-Boltzmann law), and glows most in the visible part of the electromagnetic spectrum (Wien's law -- lmax is around 500 nm). So you will see it glow with a whitish light. If you shine a flashlight on it, none of that light will be reflected to your eyes, so the object will not look any brighter than it did before you illuminated it with the flashlight.
12. lmax is 121 nm. Convert this to meters, then use Wien's law to find the surface temperature:
Light with a wavelength of 121 nm is ultraviolet light, so the object shines most brightly in the (invisible) ultraviolet. In blackbody spectra, light with progressively longer wavelengths than lmax will shine less brightly. Therefore, this object will emit violet light (400 nm) more strongly than red light (700 nm), and the overall color of the light that you see will be bluish.
14. The energy of a photon is inversely related to its wavelength. Therefore, photons with the shortest wavelengths (gamma rays) will carry the most energy, while photons with the longest wavelengths (radio waves) will carry the least. The actual formula that relates the energy of a photon to its wavelength is:
17. Different elements have different sets of electron energy-levels. Since the wavelengths of an element's spectral lines depend on how much energy it takes to move its electrons from one energy level to another, this means that different elements will display different patters of spectral lines.
18. When a source of waves is coming towards you (or alternatively, when you are going towards it), the wavelength that you measure will be smaller than the wavelength that the object is actually emitting. Conversely, when the wave-source is moving away from you (or you are moving away from it), you will detect a wavelength that is longer than what the source is truly emitting. Astronomers use this to tell if a source of light is approaching or receding, by noting whether the object's spectral lines are at shorter or longer wavelengths than they should be.
23. We want to compare the flux of Deneb (FDeneb) to the flux of the Sun (FSun), and we will need the Stefan-Boltzmann law to do this. First, write down the Stefan-Boltzmann law twice -- once for Deneb, once for the Sun:
Divide one equation by the other:
Now, knowing that the surface temperatures of Deneb and the Sun are 8,700 K and 5,800 K, respectively (see page 101 for the latter temperature), we can compare Deneb's flux to the Sun's:
This means that one square meter of Deneb's surface will emit 5.06 times more energy each second than a square meter of the Sun's surface.
26. Using Wien's law, we can find lmax for an object at a temperature of 106 K:
Converting from meters to nanometers:
This wavelength lies in the X-ray part of the electromagnetic spectrum. The moral of the story is: if you want to find a black hole, look for a bright source of X-rays. The black hole itself may be invisible, but the X-ray light from the hot stuff falling towards it can be detected.
34. You don't even need the Doppler formula to work out whether Megrez is moving towards us or away from us. The observed wavelength (l = 486.112 nm) is shorter than the true wavelength (l0 = 486.133 nm), so Megrez is moving towards us.
To get the speed, use the Doppler formula:
The minus sign means that Megrez is moving towards us (rather than away).
36. Using the Doppler formula:
That's 29% of the speed of light, or nearly 200,000,000 mph. If this cop decided to charge you $1 for every mile-per-hour over the speed limit, that would be one expensive speeding ticket!
Last edited 19 Feb 04 M. A. Weinstein.