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Chapter 11: Problem 84

Surface Temperature of Venus. Access the Active Integrated Media Module "Wien's Law" in Chapter 5 of the Universe Web site or eBook. (a) Using the Wien's Law calculator, determine Venus's approximate temperature if it emits blackbody radiation with a peak wavelength of \(3866 \mathrm{~nm}\). (b) By trial and error, find the wavelength of maximum emission for a surface temperature of \(733 \mathrm{~K}\) (for present-day Venus) and a surface temperature of \(833 \mathrm{~K}\) (as it might be in the event of a global catastrophe that released more greenhouse gases into Venus's atmosphere). In what part of the electromagnetic spectrum do these wavelengths lie?

Short Answer

Expert verified
The temperature of Venus when emitting with a peak wavelength of 3866 nm is approximately 750K. For surface temperatures of 733K and 833K, the wavelengths of maximum blackbody radiation emission are in the infrared region of the electromagnetic spectrum.

Step by step solution

01

Applying Wien's Law

Wien's law can be expressed as \(T = \frac{b}{\lambda_{max}}\) where \(T\) is the temperature in Kelvin, \(\lambda_{max}\) is the peak wavelength and \(b\) is Wien's constant, approximately \(2.9 \times 10^{-3} m.K\). In part (a), we're given \(\lambda_{max} = 3866 nm = 3.866 \times 10^{-6} m\). Substituting these values in the formula we get: \( T = \frac{2.9 \times 10^{-3} m.K}{3.866 \times 10^{-6} m} \)
02

Computing the Temperature

The result from step 1 gives the temperature in Kelvin. The computation is straightforward: \(T \approx 749.87 K\)
03

Applying Wien's Law for Given Temperatures

In part (b), we need to apply Wien's law in a reversed manner to find the peak wavelength. For a temperature of 733K and 833K, we apply the formula: \(\lambda_{max} = \frac{b}{T}\) and perform the calculations to get the peak wavelengths for these temperatures.
04

Computing the Peak Wavelengths

Using the formula from step 3, we compute the peak wavelengths for 733K and 833K: \(\lambda_{max}(733K) = \frac{2.9 \times 10^{-3} m.K}{733K}\) and \(\lambda_{max}(833K) = \frac{2.9 \times 10^{-3} m.K}{833K}\)
05

Identifying Electromagnetic Spectrum Regions

With the calculated wavelengths for the two temperatures, one to see in which part of the electromagnetic spectrum these wavelengths are. The boundaries of the infrared region are from about 700 nm (0.7 碌m) to 1 mm (1000 碌m), implying both temperatures peak in the infrared region.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Temperature of Venus
The surface temperature of Venus is a fascinating topic for scientists because it provides insights into the planet's harsh environment. Using Wien's Law, we can estimate this temperature by looking at the emission characteristics of Venus. The law helps in determining the peak wavelength of radiation emitted by a planet like Venus, which acts as a blackbody, a perfect emitter of radiation.
Applying Wien's Law,
  • The formula used is: \[ T = \frac{b}{\lambda_{max}} \] where \( T \) is the temperature in Kelvin and \( \lambda_{max} \) is the peak wavelength in meters.
  • Given that the peak wavelength \( \lambda_{max} = 3866 \) nm or \( 3.866 \times 10^{-6} \) meters, the temperature, \( T \), can be calculated.
  • By substituting the known values into this equation, we calculate the approximate surface temperature of Venus as about 750 K.
This calculation highlights the intense heat present on Venus due to its dense atmosphere laden with carbon dioxide, leading to a runaway greenhouse effect.
Blackbody Radiation
Blackbody radiation is a fundamental concept in understanding how objects like planets emit energy. A blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.
Key characteristics include:
  • It emits thermal radiation uniformly in all directions and across all wavelengths.
  • The distribution of this radiation follows a unique pattern that depends solely on the temperature of the blackbody, described by Planck's Law.
  • Wien's Law helps us find the wavelength at which this emission is at its maximum by tying it directly to the temperature of the emitting body. \[ \lambda_{max} = \frac{b}{T} \] This formula shows that as the temperature increases, the peak of the emitted spectrum shifts towards shorter wavelengths.
In the context of Venus, understanding blackbody radiation allows us to estimate its surface temperature and explore the conditions of its environment. This directly impacts how we view planetary climates and atmospheres across the solar system.
Electromagnetic Spectrum
The electromagnetic spectrum is the range of all types of electromagnetic radiation. Light is just one form of electromagnetic radiation. Others include radio waves, microwaves, infrared, ultraviolet, X-rays, and gamma rays.
Important details about the spectrum include:
  • Wavelength and frequency of radiation determine how we classify different types of electromagnetic radiation.
  • The region that includes infrared radiation is where planets like Venus emit the maximum radiation, since their surface temperatures fall within this range.
  • For Venus, with temperatures of 733 K and 833 K, the wavelengths calculated using Wien's Law fall within the infrared part of the spectrum. This means that most of the heat emitted by Venus would not be visible to the naked eye, but it can be detected using infrared sensors.
Understanding the electromagnetic spectrum is crucial as it provides the framework for interpreting and analyzing the emission lines and heat radiation from celestial bodies. This knowledge aids in the study of various astronomical phenomena and the conditions of planets like Venus.

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Most popular questions from this chapter

The Mariner 2 spacecraft detected more microwave radiation when its instruments looked at the center of Venus's disk than when it looked at the edge, or limb, of the planet. (This effect is called limb darkening.) Explain how these observations show that the microwaves are emitted by the planet's surface rather than its atmosphere.

Why is it impossible to see Mercury or Venus in the sky at midnight?

Describe the evidence that there has been recent volcanic activity (a) on Venus and (b) on Mars.

Use the Stamy Night Enthusiast \({ }^{\mathrm{TM}}\) program to observe the apparent motion of Venus on the celestial sphere. Display the entire celestial sphere (select Guides > Atlas in the Favourites menu). Open the Find pane and click the menu button in the list to the left of the label for the Sun. Select Centre from the menu that appears. Using the controls at the right-hand end of the toolbar, zoom out until the field of view is \(100^{\circ}\). Stop Time Flow and in the toolbar, set the date and time to January 1 , 2007, at 12:00:00 A.M. and the Time Flow Rate to 1 day. (a) Use the Run Time Forward and Stop time buttons to find the first date after January 1, 2007, when Venus is as far to the right of the Sun as possible, and the first date after January 1, 2007, when Venus is as far to the left of the Sun as possible. What is your interpretation of these two dates and how would you label them? (b) Set the date to December 1 , 2007 , and start the animation by clicking on the Run Time Forward button. Based on your observations, explain why Venus has neither a greatest western elongation nor a greatest eastern elongation during 2008 .

A full moon on Earth is bright enough to cast shadows. As seen from the Martian surface, would you expect a full Phobos or full Deimos to cast shadows? Why or why not?
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