OII na Teorica

u(λ, T) = 2πhc2

The Greenhouse E ect In 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth’s climate and accurately predicting the global warming caused by human industrial activities. In this problem, we will examine a simple model of global warming due to the greenhouse e ect. The greenhouse gases alter the optical properties of the Earth’s atmosphere in transmitting or absorbing Earth’s infrared radiation, resulting in a rise in the average temperature of the planet. All objects, at di erent temperatures, emit thermal radiation. The quantity indicates the thermal radiative power per unit area of an object at temperature between the wavelengths and . According to Planck’s theory of blackbody radiation, we have , (1) in which and . The wavelength corresponding to the maximum of comes from the relation (Wien’s displacement law). Indeed, using equation (1), it can be shown that , where the dimensionless quantity is the non-trivial root of an equation of the form ; you are asked to nd the function in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann law as where . Moreover, according to Kirchho ’s law of radiation, at thermal equilibrium a body absorbing a certain fraction of the incident radiation at a speci c wavelength, will radiate the same fraction of the blackbody radiation at that same wavelength. Throughout this problem assume that the Sun is a blackbody at its average surface temperature of . The Sun’s radius is and the average distance between the Earth and the Sun is . We denote by , the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. , is called the solar constant. In this problem assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then nd its numerical value accurate to three signi cant gures. The required units are indicated on the answer sheet. A. Earth as a Blackbody $u(\lambda$, $T)d\lambda$ T $\lambda$ $\lambda$ + $d\lambda$ $u(\lambda$, T) = $2\text{pihc}2$ $\lambda 5$ 1 exp( hc $\text{lambdakBT}$ ) $-1$ hc = 1. 24 $\times$ 103 eV $\text{cdotnm}$ kB = 8. 62 $\times 10-5$ eV /K $u(\lambda$, T) $\text{lambdamaxT}$ = b b = hc xmkB xm f(x) = 0 f(x) U(T) = $\text{sigmaT}$ 4 $\sigma$ = 5. 67 $\times 10-8$ W/m2K4 TS = 5. 77 $\times$ 103 K RS = 6. 96 $\times$ 108 m d = 1. 50 $\times$ 1011 m ̃uS(\lambda) S0 = \int ̃uS(\lambda)d\lambda Points: 30 Time: 5.0 Hours IPhO 2024 Theory, English (Official) Page 1 of 16

In this part, consider the Earth’s surface as a blackbody and neglect the Earth’s atmosphere. A-1 Find the solar constant, . 0.6 pt A-2 Find the Earth’s temperature, . 0.6 pt A-3 Find the function . 0.4 pt A-4 Calculate the numerical value of , and from this value , nd the value of . 0.4 pt A-5 Find for the Sun and the Earth. 0.2 pt In gure 1 the functions and are plotted versus , where is a dimensionless coe cient to rescale such that the values of the two peaks coincide. A-6 Determine . 0.8 pt S0 TE f(x) xm xm b $\text{lambdamax}$ \gamma ̃uS(\lambda) $u(\lambda$, TE) $\lambda$ $\gamma$ ̃uS(\lambda) $\gamma$ Points: 30 Time: 5.0 Hours IPhO 2024 Theory, English (Official) Page 2 of 16

Figure 1 - The plot of (red) and (blue) versus B. The Greenhouse E ect In this part, we introduce a simple model in which the Earth’s atmosphere is modeled as a thin layer at a small distance above the Earth’s surface so that the di erence between the area of the atmosphere’s layer and the area of the Earth’s surface can be neglected (see gure 2). In what follows assume that the major part of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the for each one. Also assume that the “atmosphere layer” re ects a fraction of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assume that the atmosphere does not re ect any part of the infrared radiation, however, it absorbs a fraction of the infrared radiation and transmits the rest. This behavior, known as the greenhouse e ect, changes the average temperature of the Earth. The Earth’s surface, on the other hand, re ects a fraction of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation. $u(\lambda$, TE) \gamma ̃uS(\lambda) $\lambda$ $\text{lambdamax}$ rA = 0. 255 $ϵ$ rE Points: 30 Time: 5.0 Hours IPhO 2024 Theory, English (Official) Page 3 of 16

Figure 2 - Thermal ows between the Earth and the atmosphere B-1 Assume that and , and calculate the Earth’s temperature and the atmosphere’s temperature . 1.0 pt Now assume that . In this case, the combined system of “Earth + atmosphere” re ects a di erent fraction of the solar radiation, called “albedo” and denoted by . B-2 Determine the albedo, , in terms of and . Then calculate its numerical value assuming (and ). 1.6 pt $ϵ$ = 1 rE = 0 TE TA rE $\ne 0$ $\alpha$ $\alpha$ rE rA rE = 0. 102 rA = 0. 255 Points: 30 Time: 5.0 Hours IPhO 2024 Theory, English (Official) Page 4 of 16

B-3 a) Express the Earth’s temperature in terms of , , , and . b) Using the given data and the calculated albedo, nd the numerical value of which leads to the current average temperature of for the Earth. 1.0 pt B-4 Find and determine by how much the Earth’s temperature increases if increases by one percent. 0.8 pt Assume and . These values come from real data and may di er from the results which you have obtained in the previous tasks. Now suppose that a non-radiative (e. g. convective) thermal ow is maintained from the Earth to the atmosphere, where is a constant. The quantity, , is the transmitted power per unit area. B-5 Calculate and in terms of , , , , and . 1.6 pt B-6 a) Di erentiating the equations obtained in part B-5 with respect to , nd the two algebraic equations satis ed by and . b) Use these equations to nd the numerical value of change in the Earth’s temperature as a result of a one percent increase in the value of . 1.0 pt $\sigma \alpha$ S0 $ϵ$ $ϵ$ TE = 288 K dTE $dϵ$ $ϵ$ TA = 245 K TE = 288 K JNR = k(TE $-TA$) k JNR $ϵ$ k TE TA $\sigma \alpha$ S0 $ϵ$ dTA $dϵ$ dTE $dϵ$ $ϵ$ Points: 30 Time: 5.0 Hours IPhO 2024 Theory, English (Official) Page 5 of 16

Topic: Thermodynamics, Astrophysics, Conservation of Energy Metodi: Energy Conservation Method, Physical Modeling, Photon Energy Relation, Calculus-Integration, Differential Equations Competenze: Mathematical Modeling, Physical Reasoning, Estimation & Approximation Objects: Planet, Star Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

u(λ, T) = 2πhc2

The Greenhouse Ect In 2021, Syukuro Manabe and Klaus Hasselmann were awarded half of the Nobel Prize in Physics for their work in modeling Earth’s climate and accurately predicting the global warming caused by human The Commission will examine the following: In this problem, we will examine a simple model of global warming due to the The Commission has already adopted a proposal for a regulation on the approximation of the laws of the Member States on the approximation of the laws of the Member States. The greenhouse gases alter the optical properties of the Earth’s atmosphere in transmitting or absorbing Earth’s infrared radiation, resulting in an increase in the average temperature of the planets. The objects, at their temperature, emit thermal radiation. The quantity indicates the thermal radiative power per unit area of an object at temperature between the wavelengths and . According to Planck’s theory of blackbody radiation, we have , (1) in which and . The wavelength corresponding to the maximum of comes from the relation The Commission has already adopted a number of proposals for the new rules. Indeed, using equation (1), it can be shown that , where the dimensionless quantity is the non-trivial root of an equation of the form ; you are asked to nd the function in one of the following tasks. Total radiative power per unit area of a blackbody in all wavelengths is given by the Stephan-Boltzmann Law as where . Moreover, according to Kirchho’s law of radiation, at thermal equilibrium a body absorbing a The same fraction of the incident radiation at a species c wavelength will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the incident radiation at a species c wavelength, will radiate the same fraction of the The blackbody radiation at that same wavelength. Throughout this problem assumes that the Sun is a blackbody at its average surface temperature of . The Suns radius is and the average distance between the Earth and the Sun is . We denote by , the spectral solar power radiated into a unit area of the Earth normal to the direction of radiation. The integral of this quantity over all wavelengths, i.e. , is called the solar constant. In this problem, we assume that the Earth is in thermal equilibrium and has the same temperature at all points on its surface. In all parts of the problem, express the desired quantity in parametric form in terms of the data given in the problem and then nd its numerical value accurate to three I mean, sing gures. The required units are indicated on the answer sheet. A. Earth as a Blackbody $u(\lambda$, $T)d\lambda$ T $\lambda$ $\lambda$ + $d\lambda$ $u(\lambda$, T) = $2\text{pihc}2$ $\lambda 5$ 1 Ex: hc $\text{lambdakBT}$ ) $-1$ hc = 1. 24 $\times$ 103 eV $\text{cdotnm}$ kB = 8. 62 $\times 10-5$ eV /K $u(\lambda$, T) $\text{lambdamaxT}$ = b b = hc The following is the list of the countries of the European Union: xm f(x) = 0 f(x) U(T) = $\text{sigmaT}$ 4 $\sigma$ = 5. 67 $\times 10-8$ W/m2K4 TS = 5. 77 $\times$ 103 K RS = 6. 96 $\times$ 108 m d = 1. 50 $\times$ 1011 m ̃uS(\lambda) S0 = \int ̃uS(\lambda)d\lambda The score: 30 Time: 5.0 hours The following information shall be provided: Theory, English (Official) Page 1 of 16

In this part, consider the Earth’s surface as a blackbody and neglect the Earth’s atmosphere. A-1 Find the solar constant, . 0.6 pt A-2 Find the Earths temperatures, . 0.6 pt A-3 Find the function . 0.4 pt A-4 Calculate the numerical value of , and from this value , and the value of . 0.4 pt A-5 Find For the Sun and the Earth. 0.2 pt In our 1 the functions and are plotted versus , where is a dimensionless Other

• We ‘re going to rescale . So that the values of the two peaks match. A-6 Determine . 0.8 pt S0 TE f(x) xm xm b $\text{lambdamax}$ \gamma ̃uS(\lambda) $u(\lambda$, TE) $\lambda$ $\gamma$ ̃uS(\lambda) $\gamma$ The score: 30 Time: 5.0 hours The following information shall be provided: Theory, English (Official) Page 2 of 16

Figure 1 - The plot of (red) and (blue) versus B. The Greenhouse Ect In this part, we introduce a simple model in which the Earth’s atmosphere is modeled as a thin layer at a small distance above the Earth’s surface so that the di erence between the area of the atmospheres layer and the area of the Earth’s surface can be neglected (see gure 2). In what The following assumes that the majority of the thermal radiation from the Earth and the Sun are emitted at wavelengths near the For each one. Also assuming that the atmosphere layer Re ects at fraction of the visible-ultraviolet radiation incident from above or below, and completely transmits the rest. Assumes that the atmosphere does not re ect any part of the Infrared radiation, however, it absorbs a fraction of the of infrared radiation and transmits the rest. This behavior, known as the greenhouse ect, changes the average temperature of the Earth. The Earths surface, on the other hand, re ects a fraction of the visible-ultraviolet radiation and absorbs the rest of this radiation and all the infrared radiation. $u(\lambda$, TE) \gamma ̃uS(\lambda) $\lambda$ $\text{lambdamax}$ rA = 0. 255 $ϵ$ rE The score: 30 Time: 5.0 hours The following information shall be provided: Theory, English (Official) Page 3 of 16

Figure 2 - Thermal ows between the Earth and the atmosphere B-1 Assumes that and , and calculate the Earth’s temperatures and the atmospheres temperatures . 1.0 pt Now assume that . In this case, the combined system of Earth + atmosphere re ects a of erent fraction of the solar radiation, called albedo and denoted by . B-2 Determine the albedo, , in terms of and . Then calculate its numberical value assuming (and ). 1.6 pt $ϵ$ = 1 rE = 0 TE TA rE $\ne 0$ $\alpha$ $\alpha$ rE rA rE = 0. 102 rA = 0. 255 The score: 30 Time: 5.0 hours The following information shall be provided: Theory, English (Official) Page 4 of 16

B-3 (a) Express the Earth’s temperatures in terms of , , And , and . b) Using the given data and the calculated albedo, nd the numerical value of which leads to the current average temperature of For the Earth. 1.0 pt B-4 Find and determine by how much the Earth’s temperature increases if increases by one percent. 0.8 pt He assumes and . These values come from real data and may be from the results which you have obtained in previous tasks. Now suppose that a non-radiative (e. g. Other, of a kind used for the manufacture of goods ow is maintained from the Earth to the atmosphere, where is a constant. The quantity, , is the transmitted power per unit area. B-5 Calculated and in terms of , , , , and . 1.6 pt B-6 (a) Of erentiating the equations obtained in part B-5 with respect to , nd The two algebraic equations satis and by and . b) Use these equations to nd the numerical value of change in the Earths temperature as a result of a one percent increase in the value of . 1.0 pt $\sigma \alpha$ S0 $ϵ$ $ϵ$ TE = 288 K DTE $dϵ$ $ϵ$ TA = 245 K TE = 288 K JNR = k(TE $-TA$) k The following information shall be provided: $ϵ$ k TE TA $\sigma \alpha$ S0 $ϵ$ DTA $dϵ$ DTE $dϵ$ $ϵ$ The score: 30 Time: 5.0 hours The following information shall be provided: Theory, English (Official) Page 5 of 16

Topic: Thermodynamics, Astrophysics, Conservation of Energy Metodi: Energy Conservation Method, Physical Modeling, Photon Energy Relation, Calculus-Integration, Differential Equations Competenze: Mathematical Modeling, Physical Reasoning, Estimation & Approximation Objects: Planet, Star Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)