TAP 702-6: Binary stars

Transcription

1 TAP 702-6: Binary stars Orbiting binary stars: A type of ariable star. This type of ariable star onsists of two stars orbiting around eah other. When the dier star is in front of the brighter one, the obsered intensity is at a iniu, and when the two stars are side by side, the obsered intensity is at a axiu. Relatie eloities in an orbiting binary star syste. The two stars in this question are siilar in ass, but star B is ooler and dier than star A. They orbit about their oon entre of graity X, as shown in the diagra. C A X B In one suh pair of stars, the tie for one reolution is 73 days. The stars are a distane of apart. 1 day.6 10 s 1. What are the eloities of stars A and B, relatie to the point X? 2. What is the eloity of star A, relatie to star B? The point X is not still with respet to Earth. It is oing at a eloity of 6 k in the diretion away fro Earth. 3. What are the eloities of stars A and B, relatie to Earth?. What is the eloity of either star, relatie to Earth, when it is at the position C? The Doppler shift in waelength for a star oing at a eloity relatie to the obserer is

2 λ if the speed is sall opared with the speed of light,. The speed of light,, is. 5. Find the waelength shift that an astronoer (on Earth) would obsere in a spetral line of waelength 59.0 n frotar A in the position shown in the diagra. Is this a red shift or a blue shift? 6. Find the waelength shift that an astronoer (on Earth) would obsere in a spetral line of waelength 59.0 n frotar B in the position shown in the diagra. Is this a red shift or a blue shift? 7. Find the waelength shift that an astronoer (on Earth) would obsere in a spetral line of waelength 59.0 n fro either star when it is in the position C on the diagra. Is this a red shift or a blue shift?. In any binary stars, the two stars are not perfetly lined up when seen fro Earth. This eans that there will not be any diing or brightening of the light, beause the dier star will not blok out the light fro the brighter one. How ight an astronoer tell, fro the spetru, that there are in fat two stars oing about their oon entre of ass as desribed in this question? Hints 1. You an quikly find out how far eah star goes in 73 days. Don t forget that eloity is easured in etres per seond ( ), and it has a diretion. 2. Iagine you are on Star B. How fast, and in whih diretion, does star A see to oe? 3. If the binary star is oing, it eans that the point X is oing at that eloity. You an easily obine the answers to part 1 with 6 k (taking are of units!) to get the answer here.. You need to obine 6 k with the oponent, in the diretion towards / away fro Earth, of the eloity of a star when at C. 5. This question, and the following two, follow fro the results of question 3 and. Red shifts are due to stars oing away fro Earth, blue shifts are due to stars approahing. 6. Is it approahing, or retreating? Is the waelength being strethed, or squashed?

3 7. If you e done question 6, this should be fairly obious.. Although the intensity will not hange with tie, the spetru will proided that the stars are not oing perpendiular to the line of iew. Pratial adie Fairly straightforward appliations; diffiulties arise in isualising the oeent and the agnitudes of distanes and eloities inoled. Note that soe binary stars, e.g. Capella (Aurigae) orbit in a plane perpendiular to our line of sight, and so do not show this phenoenon. The forula used for the waelength shift λ is justified due to the low speeds inoled. Note that there is a relatiisti transerse Doppler shift (a tie dilation effet) that affets the answer to question 7. Howeer, again due to the sall speeds inoled, this effet is negligible. Algol (β Persei) is the lassi elipsing syste; with a period of about 3 days. Details are gien in ost astronoy books, e.g. Norton s 2000, or irtually any enylopaedia. Or see: Alternatie approahes A battery-operated buzzer in a net bag, tied to a strong ord and whirled around the head, shows this effet quite niely. Note that the Doppler effet for sound is dependent on the otion of the ediuo that it is not stritly the sae as the Doppler effet onsidered here. Safety note: do not sak anyone around the head with the whirling buzzer! TAP 702-1: The Doppler effet Soial and huan ontext Historial perspetie: the Doppler easureents are typial of the sort of disoeries ade with the Doppler effet, one the resolution of the spetroeters was up to frations of a nanoetre. Answers and worked solutions 1. The speed of stars in orbit is equal to π D / T: πd T π 73 days ( 1 10 ) s day

4 Therefore the eloity of A relatie to X is and the eloity of B relatie to X is where the positie diretion is the diretion away fro the Earth. 2. The eloity of A relatie to B is ( ) ( ) whih is equal to with diretions defined as in question Adding the eloities in question 1 to gies Veloity of A relatie to Earth Veloity of B relatie to Earth with diretions defined as in question 1.. Here the oponent of the eloity of eah star, relatie to X, in the diretion defined in question 1, is zero, so eah star has a eloity relatie to Earth whih is the sae as that of X, i.e λ 5 ( ) ( ) n 0 This is a red shift, beause the star is oing away fro the Earth, and so the waelength is strething. λ ( ) ( ) n 1 This is a red shift for the sae reason as question λ -1 ( ) ( ) n This is also a red shift for the sae reason as question 5.

5 . When the stars are at C and the point diaetrially opposite to C respetiely, then both will hae their 59.0 npetral line red shifted by 0.13 n, as both will hae the sae eloity relatie to the Earth (question ). When they oe towards A and B, howeer, their eloities relatie to Earth inrease and derease respetiely, ausing the spetral line to be red shifted by different aounts, so it will split into two distint spetral lines. These will erge into one as the stars line up in the diretion of Earth one ore. Should the two stars be oing in a irle whose plane is perfetly perpendiular to the Earth, howeer, like the stars in the Capella binary syste, then there will be no splitting of the line, as the eloity relatie to Earth in the perpendiular diretion will not hange during the orbit. External Referenes This atiity is taken fro Adaning Physis, hapter 12, 60S