The Orbit Of Mars Big Idea Tycho Brahe Made An Observation

The Orbit of Mars Big Idea Tycho Brahe made a number of observations of the positions of Mars during the latter part of the 16th century

Tycho Brahe's meticulous astronomical observations in the late 16th century provided critical data for understanding planetary motions. Despite lacking telescopes, Brahe achieved unprecedented measurement accuracy, which Johannes Kepler later used to analyze Mars’ orbit. This experiment aims to replicate Kepler’s measurements using Brahe’s data, analyze the properties of Mars’ orbit, and investigate contemporary claims about Mars’ appearance in the night sky.

Paper For Above instruction

Tycho Brahe’s observations laid the groundwork for modern planetary astronomy, particularly through the precise positional data of Mars. His observations, taken over several years, recorded the positions of Mars relative to the fixed stars, enabling later astronomers like Johannes Kepler to deduce the orbit's characteristics. Kepler’s analysis revealed that Mars does not follow a perfect circle but an ellipse, thus challenging the long-held belief in perfect celestial circles proposed by classical astronomers. The approach involves recreating Brahe’s data triangulation method, plotting the orbital positions, calculating orbital parameters, and analyzing the nature of Mars' orbit, including eccentricity and period.

Recreating Kepler’s Measurements and Plotting Mars' Orbit

Using Brahe’s data, we begin with the known sidereal period of Mars (687 days) and the Earth’s orbital period (approximately 730 days for two orbits). The data pairs, taken 687 days apart, allow us to determine the relative positions of Earth and Mars at specific times. By plotting the heliocentric longitude of Earth and the geocentric longitude of Mars, and triangulating from the Sun-Earth and Earth-Mars data, the positions of Mars at different times can be identified. Repeating this process for five different points provides multiple positions (P1 through P5), which, when connected, approximate Mars’ orbital shape.

Plotting and Analyzing the Orbit

Using the Sun as a central point, the heliocentric Longitudes of Earth are plotted on the orbital diagram. From each Earth position, lines extend to the corresponding geocentric positions of Mars. The intersections of these lines mark Mars’ estimated positions in its orbit. The major axis of the orbit is drawn between two of these points, typically near aphelion and perihelion, as identified by Kepler. Measuring this axis provides the semi-major axis length in centimeters on the graph.

Converting Measurements and Applying Kepler’s Third Law

The scale of astronomical units (AU) is determined from the known Earth-Sun distance on the diagram. Calculating the semi-major axis in centimeters and converting it to AU involves dividing the measured length by the scale factor. This semi-major axis, when multiplied by 93 million miles per AU, yields the orbital radius in miles, which is further expressed in scientific notation for precision.

Kepler’s third law relates the semi-major axis of a planet’s orbit to its orbital period. The law states that the orbital period squared is proportional to the semi-major axis cubed. Using the calculated semi-major axis in AU, the period of Mars (in years) is derived from the formula P = √(a³), confirming the known orbital period of approximately 1.88 years.

Assessing Orbit Shape and Eccentricity

Constructing a circle with the same radius as the semi-major axis offers a baseline for a circular orbit. Comparing the actual positions P1 through P5 with this circle reveals the elliptical nature of Mars’ orbit. Eccentricity, calculated by dividing the distance from the Sun to the orbit’s focus (the midpoint of the ellipse) by the semi-major axis, indicates how elongated the orbit is. The known eccentricity of Mars’ orbit (~0.09) suggests a slightly elliptical shape, consistent with observational data.

Calculating the percent error between the measured eccentricity and the known value gauges measurement accuracy. Such discrepancies can arise from measurement inaccuracies, assumptions in plotting, or simplifications in the model.

Analyzing Mars’ Closest Approach and Mythbusting Claims

The minimum distance between Earth and Mars is derived by considering the positions of the planets at their closest approach, factoring in the orbit’s eccentricity and relative positions. Assuming Earth’s orbit is circular simplifies calculations, but in reality, Earth’s elliptical orbit also affects distance calculations. The closest pre-calculated distance provides a basis for comparing claimed observational phenomena, such as Mars’ apparent size in the sky at close approach.

The claim that Mars could appear twice as large as the full Moon (approximately 0.5 degrees in angular diameter) at its closest approach depends on its actual distance from Earth. Using the small angle formula, the angular diameter of Mars can be estimated at the closest approach. Comparing this value to the Moon’s known angular size tests the validity of the claim.

For the 2003 close approach, the actual distance of approximately 55.8 million km results in an angular diameter significantly smaller than the Moon’s. Therefore, the claim of Mars appearing twice the size of the Moon is an exaggeration, likely due to misinterpretation or optical illusion effects. This is consistent with astronomical observations and calculations, reaffirming that while Mars can appear larger during close approaches, it does not reach such extreme sizes.

In conclusion, recreating Brahe’s data using triangulation methods validates Kepler’s elliptical orbital model, illustrates the eccentricity of Mars’ orbit, and confirms that Mars’ apparent size at close approaches is substantial but not as exaggerated as claimed in the email. These exercises reinforce the importance of accurate measurement, the application of Kepler’s laws, and understanding the geometry of planetary orbits in astronomy.

References

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