IAO 2007, Практически тур, Група α — Задача 7. Radial velocity.
Practical round. Problems to solve · 29 септември 2007 г.
Note for jury and team leaders
Note for jury and team leaders. The proposed sketches are not full; the team leaders have to give more detailed explanations for students. But the correct solutions in the students’ papers (enough for 10pts) may be shorter.
Source annotations
The official O−C answer table prints calculated dates47685 and47383 for observations12 and13 respectively; these do not agree with the surrounding cycle sequence or the printed O−C values. The first interval in its final table is printed0–45878.2. All these printed entries are retained without correction.
Условие
A cepheid has its mean apparent magnitude of . Spectroscopic observations for this cepheid were carried out during 16 successive nights. The list of observations is given in the table, where the radial velocity is given in heliocentric system. Make such a plot based on the data listed in this table which enables you to derive several important characteristics of this cepheid and to answer the question: Could this cepheid be seen by the first people on the Earth (about 2 million years ago)? (The answer should be written in English: Yes or No.) What was the apparent magnitude of this star at that time?
Supplementary information: Use the relation showed in the plot below (this relation is valid for cepheids – pulsating stars).
Recommendations: Interstellar absorption and possible changes of the cepheid mean luminosity during the last 2 million years can be ignored.
| Moment of observations (in part of days) | Observed radial velocity, km/s |
|---|---|
| 0.9 | 75 |
| 1.8 | 74 |
| 2.9 | 51 |
| 3.9 | 45 |
| 4.9 | 67 |
| 5.8 | 75 |
| 6.8 | 63 |
| 7.8 | 46 |
| 8.8 | 47 |
| 8.9 | 50 |
| 9.8 | 70 |
| 10.8 | 70 |
| 11.9 | 53 |
| 12.8 | 42 |
| 13.8 | 42 |
| 14.8 | 58 |
| 15.9 | 75 |
Решение
- Use the data from table to plot heliocentric radial velocity versus time.
- Determine approximately pulsational period of cepheid from the plot ( d).
- Use the given plot to determine the mean absolute magnitude of cepheid ().
- Determine the heliocentric distance of cepheid using the known formula:
(interstellar absorption is ignored). With a given mean apparent magnitude one gets pc.
- Use the plot versus time to find approximate radial velocity of the center of mass of pulsating star (i. e. velocity of its space motion relatively to the Sun).
Thus, roughly we can estimate km/s and conclude that our Cepheids recedes from the Sun.
- During 2 million years ( y) the star passed with such a velocity the distance:
Here is the number of second in one years.
2 million years ago our star was 100 pc closer to the Sun than now, thus, its distance was at that time 700 pc. With such a distance its apparent magnitude was:
Here we neglected again the interstellar absorption and possible variations of the absolute magnitude of our cepheid during considered time interval. Thus, we conclude that 2 million year ago the star could be seen.
Оригинал в Архива: alphapract1.jpg · официални решения: practsol1.jpg · допълнителни данни: alphapract2.jpg · допълнителни данни: alphapractdata.jpg · допълнителни данни: practsol2.jpg