Energy & Telescopic Physics/Astronomy

[dekoi]

NOTE: I am only posting in this forum because nobody will answer my question in the Homework forum. Please do not move this.


I am having some difficulties with my latest assignment. I would greatly appreciate it if someone could answer the questions:

1.) Figure 2.23 on page 66 of your textbook shows the wavelengths of the first three Balmer lines: Hα, Hβ, and Hγ. Given your result in part a), what is the energy difference between the ground state and the first four excited states of a hydrogen atom?

Firstly, Figure 2.23 on page 66 has two versions of the Balmer lines of hydrogen, so which one are we supposed to use?
I would assume that we need to use the one at 320 m/s instead of 0 m/s, so that we have a value for speed when we try to find the frequency of the hydrogen.
Also, what exactly do Hα, Hβ, and Hγ represent? Are they the different excited states of the hydrogen atom?
How are we expected to compare the energy difference between the ground state and the first four excited states of a hydrogen atom? What are they represented by in the question?



2.) Popular culture often describes Mars as being inhabited by "little green men". Imagine there is a green (500 nm light), 50 cm tall Martian standing on the surface of Mars. How big a telescope would you need to observe the Martian from the Earth if your obesrvation was diffraction limited? You may assume that the Sun, Earth, and Mars are lined up when the observation is made.

I think that I have the right method of solving this problem. However, we are given a lot of space for our answer, and my answer is much shorter than all that space. On the last assignment, we were given a lot of space for question 3, and we were expected to use it all. So I'm assuming the case is the same for this assignment.

This is what I did: (Tell me if there are any errors, or anything I missed that was supposed to be taken into account).

distance = semi-major axis of Mars - semi-major axis of Earth
d = Am - Ae = 7.83 x 1010 m
λ = 500 nm = 0.5 μm
h = 0.5 m

angular resolution = height / distance = 0.5 / 7.83 x 10^10 =
angular resolution = 6.386 x 10^-12 rad

diameter = 0.25 x wavelength / angular resolution = (0.25)(0.5) / (6.386 x 10^-12)
diameter = 1.957 x 10^10 μm

Therefore, the diameter of the telescope is approximately 20,000 m.
However, this answer does not seem like enough for the amount of space we are given.


Thank you for any help you can provide.

 

[AndyW]

it is important to understand and accurately interpret data and equations. In regards to your first question, Hα, Hβ, and Hγ refer to the different energy levels of the hydrogen atom. Hα is the first excited state, Hβ is the second excited state, and Hγ is the third excited state. In order to find the energy difference between the ground state and the first four excited states, you will need to use the equation E = -13.6 eV/n^2, where n is the energy level. You can then subtract the energy of the ground state (-13.6 eV) from the energy of each excited state to find the difference.

In response to your second question, your method for finding the diameter of the telescope seems correct. However, it is important to note that the diameter of the telescope is not the same as its length. The diameter refers to the width of the lens or mirror, while the length is the distance from the lens/mirror to the eyepiece. So, the length of the telescope would be much longer than 20,000 m in order to achieve the necessary angular resolution to observe the Martian.

I hope this helps clarify your questions and provides some guidance for your assignment. Remember, as a scientist, it is important to carefully consider and analyze all data and equations to reach accurate conclusions. Best of luck with your assignment!