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Parallax happens when an object seen at two different locations appears to be at different positions. It is measured by the angle between the two lines of sight. Foreshortening, the visual effect of an object appearing shorter than it actually is due to the way it is angled towards the viewer, causes objects that are closer to the viewer to have larger parallaxes than objects that are further away from the viewer, so parallax is useful in determining distances.
Stellar parallax specifically points to when there is an apparent displacement of a star when observed in the sky at two different locations on Earth.
Parallax can be measured by the difference in position of a star as seen from the Earth at two different times, half a year apart.
If a star’s parallax is observed, using trigonometry, its distance from Earth can be calculated. The reciprocal of the parallax is the distance between the object and the observer, in parsecs.
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A parsec is the distance at which a star would have a parallax of one second of arc. 1 parsec = 3.26 light years = 3.09 x 1018 cm = 206,265 A.U.
For example, Alpha Centauri, the closest star to Earth besides the Sun, has a parallax of 0.7687. Thus, it is 1/0.7687 = 1.3009 parsecs away.
Absolute magnitude measures the intrinsic brightness of a star. Apparent magnitude measures the brightness of a star as seen by observers from Earth.
The magnitude system was first devised by the ancient Greek astronomer Hipparchus.
The brightest stars were in the first class, and the dimmest stars were in the sixth. Brighter objects have smaller magnitudes. However, later, zero and negative numbers were added to the scale.
The formula that relates the apparent (m) and absolute (M) magnitude is M = m + 5 - 5 log d, where d is the distance to the object in parsecs.
What are the distances of different stars, calculated using the reciprocal of their parallaxes, and what are the absolute magnitudes of these stars, calculated using their distances and apparent magnitudes?
The independent variable in this lab will be the stars in the sky. For each different star selected (for example, Sirius or Vega), the parallax and apparent magnitude will be different.
The parallax of a star will be measured in arcseconds, which is 1/3600th of a degree. This data is collected by the Hipparcos satellite. The resulting catalog, with data on more than 100,000 stars, can be found at: https://heasarc.gsfc.nasa.gov/W3Browse/all/hipparcos.html.
The apparent magnitude of a star has no unit associated with it. This information is also collected by the Hipparcos satellite.
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Distance = 1/parallax (arcseconds)
Distance of Sirius = 1/0.379 arcseconds = 2.639 parsecs
Absolute magnitude = apparent magnitude + 5 - 5*log distance
Absolute magnitude of Sirius = -1.46 + 5 - 5*log 2.639 parsecs = 1.43
The line-of-best-fit has a correlation coefficient of 0.7722. The correlation coefficient measures how well the line fits the data.
The distance and absolute magnitude of a star are related logarithmically, given the formula above in analysis. The correlation coefficient is 0.7627.
Conclusion and Evaluation
In this inquiry, the parallaxes and apparent magnitudes of various stars was collected. The answers to the research question “What are the distances of different stars, calculated using the reciprocal of their parallaxes, and what are the absolute magnitudes of these stars, calculated using their distances and apparent magnitudes?” are stated above in the analysis.
My results show that, in general, as the parallax increases, the apparent magnitude of a star decreases. This makes sense instinctively, since, as the parallax increases, the distance between the star and Earth decreases. If a star is closer to viewers, it will appear brighter.
The differences in the absolute magnitude calculated in this lab and the absolute magnitude determined in literature is below:
Star Absolute magnitude calculated by me Actual absolute magnitude Percentage error (%)
Sirius 1.43 1.42 0.70
Vega 0.58 0.58 0.00
Arcturus -0.33 -0.31 6.45
Altair 2.21 2.21 0.00
Alpha Centauri 4.40 4.38 0.46
Procyon A 2.62 2.68 2.24
Procyon B 3.02 13.04 76.84
Pollux 1.07 1.09 1.83
Percentage error is calculated by dividing the absolute difference between calculated results and actual results by the actual results, and multiplying the decimal by 100. For example:
(1.43-1.42)/1.42 * 100 = 0.70%
The discrepancy in absolute magnitude for Procyon B is likely because it is in a two-star system, and thus the distance between Procyon B and Earth is not the same as the distance between Procyon A and Earth.
It is not surprising that, generally, the values obtained in this lab are close to the accepted values. This is because I did not take any measurements myself, and the raw data should be extremely accurate as it was collected professionally.
The information used in this lab was obtained by the Hipparcos satellite. While it would have been nice to measure parallax by myself, I was unable to do so. Parallax can be measured independently by taking photos of the sky from the same spot six months apart, but this was not feasible for a number of reasons. First, I was not able to have half a year in which to carry out this experiment. Second, taking a clear photo of the sky requires a high quality camera, which I did not have access to. Third, the shift in position between stars is often extremely small, and this method does not lend itself to preciseness, so the differences would be difficult to determine.
Parallax can also be measured by having two different people in different locations take pictures of the same star, but again, this would require a high quality camera, and the shift between positions would still be miniscule.
In the future, it may be interesting to create a Hertzsprung-Russell diagram using this data. This would require spectral classifications of stars, which can be found online as well.
Scientific Research On Absolute Magnitude Of Stars. (2024, Feb 25). Retrieved from https://studymoose.com/scientific-research-on-absolute-magnitude-of-stars-essay
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