Estimate a lower limit for the number density of the electrons in the plasma

[blueyellow]

Homework Statement

When a returning spacecraft re-enters the atmosphere, the shock wave and the friction that it generates lead to the ionisation of gas molecules just ahead of it and around it. Because of this, during the most intense phase of deceleration, radio communication signals cannot pass between the spacecraft and the ground. If the highest frequency communications band is in the region of 450 MHz, estimate a lower limit for the number density of the electrons in the plasma that surrounds a spacecraft during radio blackout..

The Attempt at a Solution

omega(of plasma)=sqrt[((N(subscript e))e^2)/((m(subscript e)) (espsilon0))

But it seems like I can't just substitue the numbers in because there is all this about the highest frequecny and the lower limit. The highest frequency won't give the lower limit because omega^2 is proportional to N(subscript e). So what do I do? Thanks if you reply.

 

[mark726]

Thank you for bringing up this interesting question. I can help you estimate a lower limit for the number density of electrons in the plasma surrounding a spacecraft during radio blackout.

To start with, let's consider the formula you have mentioned:

ω (of plasma) = √[(Nₑe²)/(mₑε₀)]

This formula relates to the plasma frequency, which is the natural frequency at which electrons in a plasma oscillate. This frequency is directly proportional to the square root of the electron number density (Nₑ). So, as you have correctly pointed out, we cannot simply substitute the highest frequency (ω) and expect to get a lower limit for the electron number density.

However, we can use this formula to estimate a lower limit for the electron number density if we know the plasma frequency and the mass of an electron (mₑ). In this case, we can use the highest frequency (ω = 450 MHz) as the plasma frequency, assuming that it corresponds to the maximum number of electrons present in the plasma surrounding the spacecraft during radio blackout.

So, if we rearrange the formula, we get:

Nₑ ≥ (mₑε₀ω²)/e²

Now, we can substitute the values for the mass of an electron (mₑ = 9.11 x 10^-31 kg), the permittivity of free space (ε₀ = 8.85 x 10^-12 C²/Nm²), and the highest frequency (ω = 450 MHz = 4.5 x 10^8 Hz) to get a lower limit for the electron number density:

Nₑ ≥ (9.11 x 10^-31 kg)(8.85 x 10^-12 C²/Nm²)(4.5 x 10^8 Hz)²/(1.6 x 10^-19 C)²

Nₑ ≥ 3.3 x 10^12 electrons/m³

This gives us an estimate of the minimum number of electrons present in the plasma surrounding the spacecraft during radio blackout. Keep in mind that this is a lower limit, and the actual electron number density could be higher depending on various factors such as the composition of the surrounding gas, the velocity of the spacecraft, and the intensity of the shock wave and friction.

I hope this helps you in your understanding. If you have any further questions, please do