Calculating Wavelength from Radiative Power and Orbital Geometry

[vetgirl1990]

Homework Statement

The first unmanned probe will reach the stellar system of the Alfa Centauri (estimated radius 5x10⁴
km) in year 2145. The probe will enter stationary orbit of radius R 300 x10⁶km around the
star. In order to power itself the probe will convert the radiative energy received by its 10m²
antenna. It is estimated that when oriented at a right angle with respect to the incoming radiation,
the antenna will collect 30000W of total radiative power. From given information one may infer
that the λmax wavelength for which the Alfa Centauri emits the most energy, is:
a) 253nm
b) 105nm
c) 332nm
d) 78nm
e) information provided is insufficient to solve this problem

Homework Equations

E=hf
v=fλ

The Attempt at a Solution

I thought to approach this problem by first finding the amount of energy (in joules, rather than watts) that the antenna would absorb. I did this just my unit analysis:
30 000 J/s * (s / 3x10⁸m) * (3x10¹¹m) = 3x10⁷J of energy

From these equations, I isolated for wavelength:
E=hf
v=fλ
E/h = v/λ

λ = vh/E
= (3x10⁸m/s)(6.63x10^-34J/s) / (3x10⁷J)
= 6.63x10^-33m

I have a feeling that it's incorrect to apply E=hf in this situation, because that's a quantization of energy equation -- but otherwise, I don't know how to relate power to wavelength.
So this is obviously wrong. Otherwise, I don't know how to approach this problem. Any help would be great!

 

[gneill]

Hints:
1. Luminosity
2. Steffan-Boltzmann, Wien's displacement law, and black bodies

Your spacecraft is sampling a 10 m² area at its orbit radius, so you should be able to find the total luminosity of the star from that.

Can you find the star's temperature with the given information?

 

[vetgirl1990]

Hints:
1. Luminosity
2. Steffan-Boltzmann, Wien's displacement law, and black bodies

Your spacecraft is sampling a 10 m² area at its orbit radius, so you should be able to find the total luminosity of the star from that.

Can you find the star's temperature with the given information?

Ohhh, this makes so much sense, thank you! Luminosity is a new concept to me as we didn't learn the equation in class, but it can easily be applied without the equation -- just considering that light radiates from an object in a sphere, and the fact that we are given orbit radius.

How do you know when to apply Blackbody Radiation concepts (like Wein's Law, Steffan-Boltzmann) rather than general optics? My initial approach was to simply use the speed of light and wavelength relationship (v=fλ) but that didn't work.

 

[gneill]

How do you know when to apply Blackbody Radiation concepts (like Wein's Law, Steffan-Boltzmann) rather than general optics? My initial approach was to simply use the speed of light and wavelength relationship (v=fλ) but that didn't work.

I don't think there's enough information given to single out a particular frequency otherwise; some underlying mechanism has to lead you from total power to "the λmax wavelength for which the Alfa Centauri emits the most energy". When I see that sort of phrase I automatically think of black body curves with their emitted energy peaks (wavelengths or frequency versus temperature).

 

[vetgirl1990]

Hints:
1. Luminosity
2. Steffan-Boltzmann, Wien's displacement law, and black bodies

Your spacecraft is sampling a 10 m² area at its orbit radius, so you should be able to find the total luminosity of the star from that.

Can you find the star's temperature with the given information?

It seems like a lot of other students in the class were having problems with the question -- not getting one of the options listed as a MC answer. This is how our professor told us it must be solved:

" It is long three step problem. 1) you need to find the total energy/power received but the 1m2 on the orbit. (geometry) 2)you need to find the power emitted by 1m2 of the star. (geometry) 3) you find the surface temperature of the star (Stefan Bolzman Law) 4) you find the corresponding wavelength of the star ( Wien's Law) ( there is sufficient information for that)"