a)
To prove:that the perihelion distance of any planet is
It is proved that the perihelion distance of any planet is
Given information:
The polar equation:
Formula used:
Calculation:
The perihelion distance is defined as the closest distance of any planet to the sun.
So, given the fact that
Therefore
Factorize the numerator using the identity
Hence the perihelion distance is
The aphelion distance is defined as the furthest distance of any planet to the sun.
So, given the fact that
Therefore
Factorize the numerator using the identity
Hence the aphelion distance is
Therefore, it is proved that the perihelion distance of any planet is
b)
To prove: that
Given information:
The polar equation:
Formula used:
Calculation:
Substitute
Hence it is proved that
Substitute
Hence it is proved that
c)
To compute:the perihelion and aphelion distances of each planet listed in the table.
The perihelion and aphelion distances are tabulated as follows:
Planet | Semi major Axis (AU) | Eccentricity | Perihelion | Aphelion |
Mercury | 0.3871 | 0.206 | ||
Venus | 0.7233 | 0.007 | ||
Earth | 1.0000 | 0.017 | ||
Mars | 1.5237 | 0.093 | ||
Jupiter | 5.2026 | 0.048 | ||
Saturn | 9.5547 | 0.056 |
Given information:
The semimajor axes and eccentricities of the six innermost planets.
Planet | Semimajor Axis (AU) | Eccentricity |
Mercury | 0.3871 | 0.206 |
Venus | 0.7233 | 0.007 |
Earth | 1.0000 | 0.017 |
Mars | 1.5237 | 0.093 |
Jupiter | 5.2026 | 0.048 |
Saturn | 9.5547 | 0.056 |
Source: Encrenaz and Bibring The Solar System (2nd ed.) New York: Springer, p-5.
Formula used:
Formula to compute perihelion distance:
Formula to computer aphelion distance:
Calculation:
To find the perihelion and aphelion distances of Mercury:
Substitute
Substitute
Therefore, the perihelion and aphelion distances for Mercury are
To find the perihelion and aphelion distances of Venus:
Substitute
Substitute
Therefore, the perihelion and aphelion distances for Venus are
To find the perihelion and aphelion distances of Earth:
Substitute
Substitute
Therefore, the perihelion and aphelion distances for Earth are
To find the perihelion and aphelion distances of Mars:
Substitute
Substitute
Therefore, the perihelion and aphelion distances for Mars are
To find the perihelion and aphelion distances of Jupiter:
Substitute
Substitute
Therefore, the perihelion and aphelion distances for Jupiter are
To find the perihelion and aphelion distances of Saturn:
Substitute
Substitute
Therefore, the perihelion and aphelion distances for Saturn are
The perihelion and aphelion distances are tabulated as follows:
Planet | Semimajor Axis (AU) | Eccentricity | Perihelion | Aphelion |
Mercury | 0.3871 | 0.206 | ||
Venus | 0.7233 | 0.007 | ||
Earth | 1.0000 | 0.017 | ||
Mars | 1.5237 | 0.093 | ||
Jupiter | 5.2026 | 0.048 | ||
Saturn | 9.5547 | 0.056 |
d)
To find:the greatest perihelion and aphelion distances.
The planet Saturn has the greatest difference between the perihelion and aphelion and the difference is
Given information:
The polar equation:
Formula used:
Formula to compute perihelion distance:
Formula to computer aphelion distance:
Calculation:
The difference between perihelion and aphelion distances are tabulated as follows:
Planet | Perihelion | Aphelion | Difference between Perihelion and Aphelion |
Mercury | |||
Venus | |||
Earth | |||
Mars | |||
Jupiter | |||
Saturn |
The greatest difference between the perihelion and aphelion is the least value in the last column.
Therefore, the planet Saturn has the greatest difference between the perihelion and aphelion and the difference is
Chapter 8 Solutions
PRECALCULUS:GRAPHICAL,...-NASTA ED.
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning