---
title: "A model of a contracting spherical protostar treats it as an ideal blackbody radiator. The table below shows the radius R and peak emission wavelength \\(\\lambda_{\\text{max}}\\) of the protostar at three different evolutionary states.  | State | Radius | Peak Emission Wavelength | | :—: | :—: | :—: | | 1 | \\(R_0\\) | \\(\\lambda_0\\) | | 2 | \\(\\dfrac{1}{2}R_0\\) | \\(\\dfrac{1}{2}\\lambda_0\\) | | 3 | \\(\\dfrac{1}{4}R_0\\) | \\(\\dfrac{1}{2}\\lambda_0\\) |  Which of the following gives the correct proportional dependence of the total radiated power P on radius R and peak wavelength \\(\\lambda_{\\text{max}}\\), and correctly ranks the total radiated powers \\(P_1\\), \\(P_2\\), and \\(P_3\\) for the three states?"
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url: "https://nerd-notes.com/ubq/123564/"
date_modified: "2026-09-28T12:00:12+00:00"
---

# A model of a contracting spherical protostar treats it as an ideal blackbody radiator. The table below shows the radius R and peak emission wavelength \(\lambda_{\text{max}}\) of the protostar at three different evolutionary states.

| State | Radius | Peak Emission Wavelength |
| :—: | :—: | :—: |
| 1 | \(R_0\) | \(\lambda_0\) |
| 2 | \(\dfrac{1}{2}R_0\) | \(\dfrac{1}{2}\lambda_0\) |
| 3 | \(\dfrac{1}{4}R_0\) | \(\dfrac{1}{2}\lambda_0\) |

Which of the following gives the correct proportional dependence of the total radiated power P on radius R and peak wavelength \(\lambda_{\text{max}}\), and correctly ranks the total radiated powers \(P_1\), \(P_2\), and \(P_3\) for the three states?

A model of a contracting spherical protostar treats it as an ideal blackbody radiator. The table below shows the radius R and peak emission wavelength \(\lambda_{\text{max}}\) of the protostar at three different evolutionary states.

| State | Radius | Peak Emission Wavelength |
| :---: | :---: | :---: |
| 1 | \(R_0\) | \(\lambda_0\) |
| 2 | \(\dfrac{1}{2}R_0\) | \(\dfrac{1}{2}\lambda_0\) |
| 3 | \(\dfrac{1}{4}R_0\) | \(\dfrac{1}{2}\lambda_0\) |

Which of the following gives the correct proportional dependence of the total radiated power P on radius R and peak wavelength \(\lambda_{\text{max}}\), and correctly ranks the total radiated powers \(P_1\), \(P_2\), and \(P_3\) for the three states?

- **A.** \(P \propto \dfrac{R^2}{\lambda_{\text{max}}^2}\), with ranking \(P_1 = P_2 > P_3\)
- **B.** \(P \propto \dfrac{R^2}{\lambda_{\text{max}}^4}\), with ranking \(P_2 > P_1 = P_3\)
- **C.** \(P \propto R^2 \lambda_{\text{max}}^4\), with ranking \(P_1 > P_2 > P_3\)
- **D.** \(P \propto \dfrac{R^2}{\lambda_{\text{max}}^4}\), with ranking \(P_3 > P_2 > P_1\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123564/*