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586 Problem Set 1 1 due Saturday 29th of Shahrivar in class

1- Suppose a star emits radiation, but there are missing absorption lines due to the presence of some atoms in its atmosphere (the observed spectrum looks like the figure below). Suppose that there are six absorption lines as follows: λ = 99.0050 nm, 148.5074 nm, 198.0099 nm, 247.5124 nm, 396.0198 nm, and 495.0248 nm.

wavelength

(a) Which of the following four atoms do you expect to be present in the atmosphere of the star? • A with spectrum containing λ = 100 nm, 150 nm, 200 nm.

• B with spectrum containing λ = 250 nm, 400 nm, 500 nm. • C with spectrum containing λ = 600 nm, 650 nm, 700 nm. • D with spectrum containing λ = 1000 nm, 1100 nm, 1200 nm. Do you need to make assumptions about the medium between us and the star? (b) What is the velocity of the star with respect to us? Is it receding or approaching? Do you need to make assumptions about the direction of the velocity of the star? (c) Imagine that the peak of the black body distribution of the star (as observed from the Earth) appears to be at λ = 300 nm. What is the surface of the star? (d) Assuming that the observed flux of seen from the star on the Earth is known, what is the distance to the star? Do you need other information about the star? Do you need to make assumptions about the geometry of space?

2- In the rest of the course, we will be dealing with . This is a warm-up exercise. First convince yourself that among the seven SI base units, candela and mole are redundant. Furthermore, Amp`ere(or Coulomb) is introduced only because we wanted 0 to have dimensions. (We already don’t give µ0 an independent dimension in SI, why not do the 1/2 same for 0?) Omitting 0 we can express Coulomb in terms of N m via Coulomb’s law. So we will work in the so-called “rationalized Lorentz-Heaviside” units in which Maxwell’s equations are precisely those of SI units except that both 0 and µ0 are set to 1: ∇ · E = ρ, ∇ · B = 0, ∇ × E = −∂tB, ∇ × B = J + ∂tE. (1) (a) Left with only four dimensions (, , , temperature), we can choose appropriate units (which we call natural units) in which c =h ¯ = kB = 1 are dimensionless. Show that all quantities can be expressed in terms of a unit of energy (or powers of it). It is conventional in to adopt GeV as the unit of energy. Express the following quantities in terms of GeV (or powers of it): 1 meter, 1 , 1 kilogram, 1 Kelvin and e (electron charge). −1/2 (b) What is ’s G in terms of GeV? Define the “reduced Planck mass” Mp = (8πG) 1 and write the quantities of part (a) in terms of Mp (or powers of it) instead of GeV. Argue that one can abandon using units altogether by writing everything in “” (the ultimate natural units).

1 −1/2 It is also conventional to work with the “Planck mass” MP = G , but in this course we use the reduced Planck mass.

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