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2

You need to look in to the GZK cutoff. In 1966, Greisen, Kuzmin and Zatsepin calculated that above a threshold of $5\times10^{19} eV$ cosmic ray protons would lose energy to photo-pion production on the cosmic microwave background fairly rapidly. The consequence of this is that cosmic rays above that energy can't travel more than about $50 Mpc$ without ...

2

Hints: Note that the derivative of the sign function $${\rm sgn}^{\prime}(z)~=~2\delta(z) \tag{A}$$ is twice the Dirac delta distribution. This fact seems to be at the heart of OP's question. Repeated differentiations of the Mestel disk potential $$\Phi~:=~ v_0^2 \ln(r+|z|), \qquad r~:=~\sqrt{R^2+z^2}, \tag{B}$$ leads to \frac{\partial \Phi}{\partial ...

3

I'll add a few more options for getting the ages of stars, beyond the HR diagram technique mentioned in Chris White's answer. If you can get a R=50,000 optical spectrum of a star with decent signal to noise ratio will quite easily give you the temperature (to 100K), surface gravity (to 0.1 dex) and metallicity (to 0.05 dex), plus a host of other elemental ...

2

The necessity of GR to solving a particular problem can be assessed by calculating $GM/Rc^2$. Here, $M$ is the mass involved in producing a gravitational field at some separation $R$. The rule of thumb is that if $(GM/Rc^2)\ll 1$ (i.e. is close to zero) then GR effects (time dilation) can be neglected roughly at that order of precision. So if we take an ...

0

Hopefully an observer can chime in with better advice... but: Generally, people calculate the (average) brightness in rings around the center, then plot that brightness as a function of the projected radius (i.e. angle, e.g. in arcseconds). You can check to make sure your solid angle is correct (for example you say $\delta$ is the complementary angle to ...

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