Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Lecture Notes

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia To develop a simple model of Earth’s Climate. To develop a model of the greenhouse effect. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Model: a simplified and idealized physical and mathematical construct that allows one to understand and make useful predictions about a real system. Steady-state: mean power coming in (P in ) must equal the mean power going out (P out ), all the time. Thus Earth’s temperature is constant (~14°C). Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The Earth is a closed thermodynamic system, freely exchanging energy with the rest of the universe, but not matter (except for tiny amounts). The Earth is a vacuum thus energy is lost in the form of radiation. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Black-body radiation: an object’s temperature determines at what rate radiation is emitted, and at what wavelengths. A black body is an idealized object that is a perfect absorber as well as a perfect emitter of electromagnetic (EM) radiation. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Climate Model Figure 1. The electromagnetic spectrum with corresponding temperatures of radiation emitting bodies.

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Methods of energy transfer by radiation: 1.Transmission: It can pass through the object. ie. A window. 2.Reflection: emission from a surface. ie. A mirror. 3.Absorption: The radiation is retained within the object it hits. The object will then emit energy as black body radiation depending on its temperature. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The wavelength of emitted radiation depends on the temperature of the black body object. The temperature of a black body depends on the percentage of radiation that is absorbed and re-emitted. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The energy of EM radiation that is emitted or absorbed by an object depends mainly on its temperature, as shown by the Stefan-Boltzmann’s Law: P = σAεT 4 P is the power radiated, or the amount of energy per second (units: Watts, W) σ is the Stefan-Boltzmann constant, equal to x10 -8 W/m 2 ∙K 4 A is the area of emission (units: square metres, m 2 ) ε is the emissivity of the object, or the fraction of EM radiation a surface absorbs (0≤ ε ≤1) T is the temperature of the object (units: Kelvins, K) Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia How much power does the Sun radiate onto Earth? Sunlight, or solar radiation, includes the total spectrum of electromagnetic radiation given off by the Sun. This solar radiation is emitted in a spherical distribution. No solar power is absorbed by interplanetary space (a vacuum). Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Figure 2. The solar radiation, emitted by the Sun in a spherically symmetric distribution, coming into contact with Earth. Image not to scale. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia thus, pinhead, basketball, 29.2 metres between Climate Model Image 1. A scale image of the Earth in relation to the Sun. The Earth is represented by the white pinhead and the Sun by a basketball. The two are 29.2 m apart, approximately the length of a basketball court.

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Image 2A. A scale image of the Earth represented by a white pinhead. Climate Model Image 2B. A scale image of the Sun represented by a basketball.

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The relative size of Earth is incredibly tiny in relation to the Sun It can be approximated that the ratio of its projected 2D area on the 3D surface area of the solar radiation distribution is equal to the fraction, f, of the solar power incident on the Earth. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Figure 3. The projected area of Earth on the spherically distributed solar radiation emitted by the Sun. Above, Earth is a disk with a radius, r e, of 6.37x10 6 km. The dashed lines indicate the slice of incident solar radiation on Earth. Image not to scale. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Using the Stefan-Boltzmann Law, and assuming the Sun is a black body (ε = 1) P s = 4πr s 2 σT s 4 P s ≈ 3.9x10 26 W Thus, Earth’s incident solar power can be found as P e = f ∙ P s P e ≈ 1.77x10 17 W Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia A fraction of solar radiation is reflected straight back into space without ever warming the Earth. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia This reflective property is called the albedo, A. For Earth, A≈0.3, and is mainly due to clouds, haze and ice. Therefore, Earth’s incident power must have a correction term, where P in = (1 – A) ∙ P e P in ≈ 1.23x10 17 W Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The incident solar radiation, S, on the surface of Earth’s atmosphere that the sunlight shines on is Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The mean incident solar intensity, I in, on the entire surface of Earth as averaged over the entire year is: Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia How much power does Earth radiate? The power emitted by Earth is P out = 4πr e 2 σT e 4 where the Earth is assumed to be a black body, so ε = 1. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The solar intensity emitted from Earth’s surface is Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The simple model so far assumes that Earth lacks an atmosphere. Earth’s atmosphere is mostly transparent to solar radiation (44% visible, 52% near infrared (IR), 4% ultraviolet (UV)). Therefore, most of Earth’s incident solar radiation comes through the atmosphere and warms us. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Earth’s atmosphere also absorbs much of its own radiation (longer wavelength IR). The atmosphere acts like one way glass, allowing solar radiation to enter, but preventing the Earth’s radiation from exiting. This is called the Greenhouse Effect because glass behaves in a similar fashion. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Did you know... We can see through windows because our eyes absorb visible light. If, however, we were looking through infrared lenses, a window would appear to be a mirror. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Climate Model Image 3. The image on the left is taken with a regular camera and illustrates the properties of visible light. The image on the right is taken with an infrared camera and shows the windows emitting infrared radiation (in the form of hear) and illustrate that they are no longer appear transparent.

To incorporate the greenhouse effect into our simple model let’s make the following assumptions: there is only one layer of Earth’s atmosphere. the atmosphere allows most of the incident solar radiation through, but absorbs radiation emitted by Earth. the atmosphere then radiates equally from both its topside and underside. Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Climate Model

The equation for the conservation of energy on Earth’s surface is The equation for the conservation of energy of Earth’s atmosphere becomes Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Figure 4. A diagram of the exchange of EM ra diation between the Sun, Earth, and Earth’s atmosphere. The green arrows represent the incident solar intensity, which is not absorbed by Earth’s atmosphere. The red arrows represent IR radiation. The red equations represent the mean solar intensity, I in or I out, where ℰ = 1. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The temperature implications of this model are as follows: EARTH’S ATMOSPHERE SURFACE EARTH’S SURFACE I in = 240 W/m 2 I in = 480 W/m 2 I in = I out = σT a 4 I in = I out = σT e 4 T a = 255 k = -18°C T a = 303 k = 30°C Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia This temperature for Earth’s surface is much too hot! Earth’s mean surface temperature is recorded as a mean of 14.5°C. This model assumes a single but perfect greenhouse layer, which in reality is not accurate. In reality, there are many factors that contribute to this difference. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia To improve our model, we will focus on the first of these factors. There are holes in our atmosphere, so Earth’s atmosphere only absorbs a fraction of the IR radiation that Earth emits. In other words, ℰ ≠ 1, but ℰ = 0.9, the emissivity of air. Therefore, an observer in space would detect IR radiation emitted by Earth’s surface as well as Earth’s atmosphere. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia The equation for the conservation of energy on Earth’s surface is now The equation for the conservation of energy of Earth’s atmosphere becomes Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia Figure 5. A diagram of the exchange of EM radiation between the Sun, Earth, and Earth’s atmosphere. The green arrows represent the incident solar intensity. The red arrows represent IR radiation. The red equations represent the mean solar intensity, I in or I out, where ℰ =0.9. Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia From this data, the temperature implications are as follows: Therefore, this corrected model produces a mean temperature for Earth’s surface that is very close to the measured mean temperature of 14.5°C. EARTH’S ATMOSPHERE SURFACE EARTH’S SURFACE T a = K = -30.1°C T a = K = 15.8°C Climate Model

Physics and Astronomy Outreach Program at the University of British Columbia Physics and Astronomy Outreach Program at the University of British Columbia 1.SOHO/Extreme Ultraviolet Imaging Telescope (EIT) consortium. Visual Tour of the Solar System: The Sun (online). About.com. [May 5, 2009]. 2.NASA. Electromagnetic spectrum (online). netic%20spectrum [May 19, 2009] netic%20spectrum 3.NASA/Goddard Space Flight Center, Scientific Visualization Studio. Apollo th Anniversary: Saudi Arabia (online). Nasa. [May 4, 2009]. 4.Çengel, Yunus A. Steady Heat Conduction. In: Heat Transfer a Practical Approach (2). New York: McGraw Hill Professional, 2003, p Climate Model