USAAAO 2014, Национална олимпиада (NAO) — Задача 4. Section B: Long Questions, 3
National Astronomy Olympiad 2013-2014 · 10 т.
Указания
Duration: 2 hours 30 minutes
Maximum Points: 66 (Multiple Choice: 32 points, Long Questions: 34 points)
This test is designed to be taken with an answer sheet and scantron sheet on which the student records his or her responses. All answers are to be marked on these sheets, not written in the booklet. Anything written in the test booklet will not be scored. Students must begin each Long Answer on a new sheet, clearly indicating the question number. Each student should be provided with an answer sheet and scratch paper, both of which must be turned in with the test booklet at the end of the examination.
The full examination consists of 16 multiple-choice questions and 3 Long Answer questions representing a fairly wide range of difficulty. Students should be permitted to use scientific calculators. A table of physical constants other useful information are provided as an attached booklet.
Do not discuss this examination with anyone after its completion. Your results will be emailed to you by March 20th.
Declaration
I hereby affirm that all work on this exam is mine, and that I took this test under a proctor's supervision, with no outside aids beyond the materials provided and allowed. Furthermore, I also affirm to not discuss the test with others or provide any sort of aid to other examiners through the course of the examination. I understand that failure to do so may result in your disqualification from the exam.
Signature: ____________ Date: ____________
Long Answers
Please note that there are other possible methods of solving these questions. Points were awarded for any method with correct reasoning. Points were also given for partially correct solutions and errors carried forward, however one point was deducted for an incorrect final answer.
Условие
The critical density of the Universe is the density at which the gravitational attraction of matter within the universe is balanced with its expansion in such a way that neither will ultimately prevail. If the density of the universe were lower than this critical density, expansion will continue indefinitely. On the other hand, if the density of the universe were any higher, it would re-collapse upon itself.
(a) The critical density of the universe is given by:
It is possible to arrive at this result through classical means. By modeling the universe as of having infinite extent and uniform density, expanding linearly with velocity derive the above expression. [6] [6 т.]
(b) The density of ordinary (baryonic) matter in the universe is determined to be about . Hence, provide a justification for the existence of dark matter. (Latest estimates of the Hubble constant suggest a value of ) [4] [4 т.]
Отговори
- (b) Official solution: rhoC = 8.66 × 10⁻²⁷ kg m⁻³, much larger than the baryonic density, so the remaining mass must be dark matter.
Решение
a)
Consider a particle of mass at a distance from the center.
Kinetic Energy = [2]
Gravitational Potential Energy = [2]
For the particle to be critically bound:
Kinetic Energy + Gravitational Potential Energy = 0 [1]
[1]
b)
- [1]
- The density of the universe must be close to [1]
- As predicted by inflationary models/measurements of universe curvature/other reasonable explanation [1]
- Since , the remaining mass must be made up for by dark matter. [1]
Оригинал в Архива: 2014-nao-exam.pdf · официални решения: 2014-nao-solutions.pdf