In the times before Galileo, many people believed that the solar system was Earth-centered. The Geocentric model of the Earth had one problem as discovered by Galileo: it could not explain the phases of the planet Venus. Because Venus is an inner planet between the Earth and the Sun, depending on where Venus is in its orbit, we see different phases similar to those shown by the moon. These phase changes cause the planet not to be a constant light source, which affects its % Illumination. Also, the change in distance leads to a change in angular size and together they affect the magnitude.
Distance V. Brightness
Distance and Brightness are closely related. In fact, they have an inverse square relation. As the distance of a constant object increases, the square of the brightness decreases. That is, Distance= 1/ (Brightness)^2. This Inverse Square Law holds for constant light sources.
Brightness of Stars
The Greek Astronomer Hipparchus (160-127 BC) divided the stars into six different classes. The brightest he called first-class stars, and the next fainter second-class stars, until finally with the most faint stars being the sixth-class. The larger the magnitude number, the fainter the star. Astronomers have adopted Hipparchus' system, but had to revise it to include the very faint and very bright stars. Astronomers now have telescopes that reveal many more stars than what the human eye can detect, such as the Hubble Telescope which can detect stars as faint as 28+ magnitude. Astronomers also extended the scale into negative numbers to include brighter objects. For example, Vega is almost a zero magnitude at 0.04, and Sirius, the brightest star in the sky, has a magnitude of -1.42. The sun can be placed on the scale at -26.5 and the moon at -12.5 (See Figure 1).
These numbers are known as apparent visual magnitudes and describe how the stars look to the human eye from the earth. A star that is a million times more luminous than the sun might appear very faint if it is far away, and a star that is much less luminous might look bright if it is nearby. Apparent visual magnitudes tell us only how bright the stars appear in our sky.
Magnitudes
Although Astronomers have adopted the Greek magnitude scale, measurements require that the scale be given mathematical precision. The magnitude system that Hipparchus devised is based on a constant intensity ratio of about 2.5 for each magnitude (Intensity being a measure of the light energy from a star that hits 1 square meter in 1 second). If they differ by 2 magnitudes, their ratio is 2.5X2.5, and so on.
When 19th-century astronomers began measuring starlight, they needed to define a mathematical magnitude system that was precise, but they wanted it to agree roughly with Hipparchus. The stars that Hipparchus classified as first and sixth differed by 5 magnitudes and have an intensity ratio of almost exactly 100, so the modern system of magnitudes specifies that a magnitude difference of 5 magnitudes corresponds to an intensity ratio of 100. That means that 1 magnitude corresponds to an intensity ratio of precisely 2.512, the fifth root of 100. That is, 100= (2.512)^5. From this formula, we see that the intensity of a star differs by a power of 100 for every 5 magnitudes increase (See Figure 2).
Venus and Its Phases
In the times before Galileo, many people believed that the world was Earth-centered. The Geocentric model of the Earth had one problem as discovered by Galileo: it could not explain the phases of the planet Venus. Because Venus is an inner planet between the Earth and the Sun, depending on where the Venus is in its orbit, we see different phases similar to those shown by the moon. These phase changes cause the planet to be a variable light source as its % Illumination changes. Also, the change in Distance produces a change in Angular Size and together they affect the Magnitude.
Purpose:
To explain how the apparent visual magnitude of the planet Venus depends on its distance from the Earth, and to devise a mathematical formula relating magnitude and distance.
Hypothesis:
When the planet Venus is farther away from the Earth it will appear dimmer (and have a larger magnitude) than if it were closer it would become brighter (with a small magnitude value), an follow the Inverse Square Law: Brightness= 1/(Distance)^2. By gathering actual distance and magnitude data for Venus over a long period of time, then analyzing this information graphically, it should be possible to derive a formula which describes the mathematical relationship between magnitude and distance.
Method:
1) Obtain a copy of the Redshift Planetarium Software. Using this program, obtain information on the planet Venus. Information needed is: Magnitude, Earth/Venus Distance, and % Illumination. Record this information for every other day of the year.
2) The information obtained should be for half of Venus' orbit. You can tell this when the % Illumination goes from 100% to 0% or visa versa.
3) Take the information and using a Microsoft Excel Software Program put the data on a spreadsheet.
4) On the Excel program, calculate the Angular Size by using the Small Angle Formula:
q=(Diameter of Venus X 206265(")) / (Earth-Venus Distance(AU))
5) To find Relative Brightness, find the maximum value of magnitude and use this as your constant. Plug this into the formula:
Relative Brightness= 2.512^(max. value-magnitude)
6) Using the Graphical Analysis Software Program, make the following graphs:
Magnitude V. Distance
% Illumination V. Distance
Angular Size V. Distance
Relative Brightness V. Distance
Magnitude V. % Illumination
7) Using the Automatic Curve Fit mode, try and find a mathematical relationship for the graph.
8) After a close curve fit has been found, test the formula. To do this, use Microsoft Excel. With your new formula calculate the new values.
9) Compare these to the given values by finding Absolute Error using:
Absolute Error = (Calculated Value - Given Value)
10) Repeat steps 1-4 for each of the graphs until a mathematical formula has been found for each.
Graphs:
The graph above shows how Venus' angular size decreases with its distance from Earth. Compare with the graph below, which documents the increasing fraction of the planet's disk that is visible from Earth as the separation distance increases
The graph above shows that Venus is about 25-30% illuminated when it reaches maximum brightness. (Remember, lower magnitude number corresponds to a brighter object.) Venus actually appears dimmest when it is 100% illuminated!
Conclusion:
I found that for the planet Venus the point in its orbit when Venus is .42 AU away from the Earth is where the brightest magnitude occurs, -4.62. The following formulas relating Venus' visual characteristics were successfully derived:
Magnitude=A+ B*x+ C*x^2+ D*x^3+ E*x^4+ F*x^5+ G*x^6+ H*x^7
-(where A=8.5853; B=-108.90; C=361.88; D=-634.33; E=640.99; F=-374.62; G=117.57; H=-15.333; x= Distance in AU).
% Illumination=A*ln(B*x)
-(where A=52.5; B=3.80; x=Distance in AU).
Angular Size=A/x
-(where A=17.6; X=Distance in AU).
Relative Brightness=A+B*x+C*x^2+D*x^3+E*x^4+F*x^5+G*x^6+H*x^7
-(where A=-16.503; B=150.58; C=-494.54; D=855.63;
E=-854.03; F=493.95; G=-153.72; H=19.912; x=Distance in AU).
Magnitude=A+B*x+C*x^2+D*x^3+E*x^4+F*x^5+G*x^6+H*x^7
-(where A=-3.9854; B=-0.082113; C=0.0044672;
D=-0.00013795; E=2.6163e-6; F=-2.8215e-8; G=1.5790e-10;
H=-3.5872e-13; x=% Illumination).
Data Discussion
I found part of my hypothesis to be incorrect. When the planet Venus is at a distance of .42 AU away from the Earth, it reaches its brightest magnitude, -4.62. This position is neither the closest nor the farthest distance from Earth. When the distance gets smaller or larger than .42 AU, then Venus actually gets dimmer.
After I studied the % Illumination V. Distance for Venus, I found that as Venus gets farther from Earth its % Illumination gets bigger. After the distance becomes greater than .42 AU, the effect of increasing % illumination is overshadowed by the effect of increasing distance. As Venus becomes more distant its angular size decreases, producing a counteracting dimming effect. Overall, the magnitude gets larger (or in other words, Venus becomes dimmer). As Venus gets closer to Earth, approaching .42 AU, the % Illumination goes down, but because it is closer, the magnitude value gets bigger. Magnitude is affected by % illumination and magnitude. All three depend on distance.
The formulas for Magnitude V. Distance and Relative Brightness V. Distance graphs both show a polynomial function to the 7th power.
By analyzing the % Illumination V. Distance data, I found that for Venus,
% Illumination = 52.5*ln(3.80*Distance) with Distance in AU. This formula was verified using Microsoft Excel.
An analysis of Angular Size V. Distance was also done. I found that as Venus got farther from the Earth, the Angular Size got smaller. The formula this follows is: Angular Size (") = 17.6/Distance (AU). This formula was also checked using Excel.
The Magnitude V. % Illumination is a polynomial function to the 7th power. If an observer is looking at Venus and notices the phase (% Illumination), by using the formula found and verified, the observer should be able to calculate the apparent visual magnitude.
All derived formulas were tested by entering observed data, calculating each result, and then comparing the outcome to the actual known value. In every case the results were extremely successful.
Further Studies:
I have enjoyed this project, and see great potential for more research and analysis. Since the information analyzed was for only half of Venus' orbit, a study on a full cycle would be more complete.
Mercury is also an inner planet. It would be interesting to see if it also followed the same formula of a polynomial to the 7th power as Venus does.
Outer planets do not show phases as inner planets like Venus. I would like to find out if they followed the Inverse Squared Law, or have their own unique formulas.
Acknowledgements
I would like to give a special thanks to the following individuals for helping me throughout my project: Mr. Sweet, Mr. McCarroll, Mr. Beals, and my parents Duane and Malinda Black.
Bibliography
Seeds, Michael A. Foundations of Astronomy. Wadsworth Publishing Company, California: 1997 pp. 12-14.