Solving for the Energy and Direction of Emitted Photons from a Moving Pion

[mike217]

The neutral pion is an elementary particle of the meson family that has a rest mass energy of 135 MeV. This particle is unstable and decays into two photons ("light particles" of no rest mass and energy E=pc). Consider now the following situation: a neutral pion has a kinetic energy of 270 MeV as measured in a given frame. Find the momentum, the direction of propagation and the energy of each of the two emitted photons, given that one of the photons is emitted in a direction perpendicular to the initial velocity of the neutral pion.

My solution: by conservation of energy Kpion+Mpion*c^2=(p1+p2)c^2
p1+p2=(270+135)/c=405/c

by conservation of momentum:

x-direction: Ppion=p1cos(theta1) + p2cos(theta2)=p2cos(theta2)
y-direction: 0=p1-p2sin(theta2)

For the pion we have Ppion*c^2=(E^2-mc^2)^0.5=4.498E-15*c^2

My problem now is solving for theta2. theta2=arctan(P2y/P2x)
I already know P2x=Ppion, but I can't get P2y.

Thanks.

 

[dextercioby]

You have a system of 3 equations with 3 unknowns
$$p_{2}\cos \vartheta=P$$ (1)

$$p_{2}\sin\theta=p_{1}$$ (2)

$$p_{1}+p_{2}=\frac{K}{c}+Mc$$ (3)

,where M is the rest mass of the neutral pion,K is the KE of the pion & P is the pion's (relativistic) momentum...

Solve the system & find the 3 unknowns...

Daniel.

 

[mike217]

Hi Daniel I am getting the following,

from (1): p2=P/cos(theta2)
from (2): p1=p2sin(theta2)

substitution in (3) yields p2sin(theta2)+P/cos(theta2)=K/c+Mc
and after a few manipulations I get tan(theta2)+
sec(theta2)=(K/c+Mc)/P I am not sure how to solve
for theta2.

 

[Curious3141]

Hint : take equation (1) squared plus equation (2) squared.

 

[dextercioby]

That doesn't work...The "theta's" are different... Cf. $\theta$ to $\vartheta$.

It was a joke...They're the same.

Good advice...

Daniel.

 

[mike217]

That doesn't work...The "theta's" are different... Cf. $\theta$ to $\vartheta$.

lol that's hillarious.

Thanks to all for their kind help.

 

[Curious3141]

That doesn't work...The "theta's" are different... Cf. $\theta$ to $\vartheta$.

It was a joke...They're the same.

Good advice...

Daniel.

Were the "different" thetas a typo ?

Anyway, let me just work this out as I would do it.

Using Daniel's notation,

Square (1) and (2) and add :

$$p^2_2 = p^2_1 + P^2$$


$$p^2_2 - p^2_1 = P^2$$


$$(p_2 - p_1)(p_2 + p_1) = P^2$$ ---(4)

Put equation (3) into (4) and rearrange,

$$p_2 - p_1 = \frac{P^2}{\frac{K}{c} + Mc}$$ ---eqn(5)

Take (3) + (5) :

$$2p_2 = \frac{P^2}{\frac{K}{c} + Mc} + \frac{K}{c} + Mc$$

and you can find $p_2$ and then $p_1$

How to get $P^2$ in terms of what's given in the question ? For that, I would use $$E^2 = P^2c^2 + m^2c^4$$

where $$E = K + mc^2$$. You're given $K$ and the rest energy.

I'll leave the orig. poster to do the final simplifications.