Astronomy Olympiad
Problems on Physical Properties of Stars
Parallax
_{}
By comparing two photographs taken exactly six months apart the angle of parallax (p) can be determined in arc-seconds. Hence the distance to the star can be found as d in parsec.
Pogson’s Equation
_{}
The apparent magnitude of any star (m) can be related to its apparent brightness (B) by Pogson’s equation, where k is an experimental benchmark.
Inverse square law for Brightness
_{}
Brightness reduces with distance as an inverse square of distance. Thus brightness of an object as observed from two distances r_{1} & r_{2} is as shown in the formula.
Absolute Magnitude
_{}
Absolute magnitude is defined as the magnitude of the object when it is observed from a distance of 10 parsec. Using the 1/r^{2} relationship we get the above. Absolute magnitude can also be determined by measuring the parallax of the object.
Distance Modulus
_{}
The LHS is called as the distance modulus. If m is measured and M is determined by astrophysical reasoning, then distance D can be found in parsec.
Corrections
_{}
The earth’s atmosphere as well as the intervening medium between stars or galaxies (usually called the Inter-stellar medium), dim the starlight by absorbing it. Hence the magnitude measurements have to be corrected by a certain factor A for each absorption. A_{ISM} is determined experimentally by observing many stars with similar M (judged on the basis of their physical properties) and correlating their magnitudes. A_{atmosphere} can be determined by measuring the magnitude of the same star at various points in the sky, since the density of the layers of the atmosphere and the distance through which starlight travels in the atmosphere, play a role here. a is the extinction coefficient for the medium which makes up the atmosphere.
Luminosity
_{}
Luminosity of a star can be obtained by measuring the bolometric absolute magnitudes (measured by using instruments equally sensitive to all wavelengths) of the star and the sun. It is convenient to express luminosities with respect to solar luminosity (3.79 x 10^{26} J/s). By Stefan’s law the energy density of a star having effective spectral temperature T_{eff} is given as_{}. Thus Luminosity, which is the net energy given by the entire surface is_{}. The energy density received from the star at a distance r is
_{},
where _{} is the angular diameter of the star as viewed at a distance D. The energy density as measured above for the sun from the earth is called the solar constant. If we know the bolometric apparent magnitude of the star, its angular diameter and its effective temperature, we can use Pogson’s equation to obtain
_{}
Magnitudes of Star Systems
_{}
Apparent brightness of a system of stars such as a cluster or a galaxy can be found by adding up the intrinsic brightness (luminosities) of the various components (stars). Hence, using Pogson’s equation one can calculate the magnitudes. It is often easier to express the brightness in terms of solar brightness.
_{}
Problems
- What is the farthest distance a telescope a telescope can measure using trigonometric parallax if its resolving power is 15 m-arcsec? Assume that the stars are point sources. Why can’t you measure the distance to the Andromeda galaxy using the technique of parallax?
- The magnitudes of a star are measured using two photometers A & B, and are found to be 2.56^{m} and 2.54^{m} respectively. If A is known to be an accurate instrument, what is the error in B? If you did not know which of the instruments was faulty then what would you take as the magnitude of the star?
- What would be the brightness of a star (m = -1.4^{m}) if the star were brought to a third of its distance, Rigel Kentaurus is taken as the standard reference star with magnitude -0.01^{m}?
- Fill in the “???” marked gaps in the table below. What do you observe from the table?
HIP | Common name | m_{vis} | M_{vis} | D (pc) | Spectral Type |
53020 | EE Leo or Wolf 358 | ??? | 12.89 | 5.6351 | M4: |
25878 | Wolf 1453 | 7.97 | ??? | 5.6909 | M1V |
26857 | Ross 47 | 11.56 | 12.75 | ??? | M5 |
86990 | | 10.75 | 11.93 | 5.8113 | M5 |
- Year 2050, Earth has been destroyed by a meteoric impact. You survived on a ship circling Mars You need to send a radio signal for help, to all the known galactic neighbors. Your ship may not survive or more than 30 years. Which of them would be able to come to your help? (You can use the data that was transmitted to you earlier)
Civilization | Planet | Star | m_{vis} | M_{vis} | p (mas) |
Bellaris | Bellar | Bellatrix | 1.64 | -2.72 | |
Arcturamans | 3-Arcturi | Arcturus | -0.05 | | 88.89 |
429/25 | 429 | Fomalhaut | 1.17 | 1.74 | 130.03 |
Vegnolans | Vegnola | Vega | 0.03 | 0.58 | |
- The following stars were observed and corrected for the Earth’s atmosphere. Can you determine the magnitude of the ISM?
Common Name | m_{vis}_{} | M_{vis}_{} | D |
Shaula | 1.62 | -5.05 | 215.52 |
El Nath | 1.65 | -1.37 | 40.18 |
Alnilam | 1.69 | -6.38 | 411.52 |
Al Nair | 1.73 | -0.73 | 31.09 |
Alnitak | 1.74 | -5.26 | 250.63 |
Alsuhail | 1.75 | -5.31 | 257.73 |
- What is the value of the solar constant when measured just above the Earth’s atmosphere?
- Estimate the apparent magnitude of the Andromeda galaxy?
- Consider a binary star system with component stars A & B with apparent magnitudes -1.05^{m} and 2.51^{m}, respectively, at a distance of 55pc. What is the combined apparent magnitude of the system?
- Consider a globular cluster of n stars each of apparent magnitude m. Find the total apparent magnitude of the cluster.