Example: Finding the Equation of an Elliptical Orbit Given Distances of AU and AU
Consider a planet moving in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately AU and the furthest is approximately AU. By centering the ellipse at the origin as shown in the corresponding graph, the vertices are located at and , which means and . The sun is located at a focus at the point , meaning . To find the denominator for the term, use the relationship . Substituting the known values gives . Substituting and into the standard form equation for an ellipse centered at the origin yields the final equation: .
0
1
Tags
OpenStax
Intermediate Algebra @ OpenStax
Ch.11 Conics - Intermediate Algebra @ OpenStax
Algebra
Related
Example: Finding the Equation of Pluto's Elliptical Orbit
Example: Finding the Equation of an Elliptical Orbit Given Distances of AU and AU
Example: Finding the Equation of an Elliptical Orbit Given Distances of AU and AU
As a computer-aided drafting (CAD) technician, you are documenting the standard procedure for extracting the mathematical equation of an elliptical gear directly from its 2D grid profile. Arrange the procedural steps to find the equation of an ellipse centered at the origin from its visual graph in the correct sequential order.
A landscape architect is designing an elliptical garden bed and needs to document its mathematical equation. The blueprint shows the garden centered at the origin of a coordinate grid. According to the standard procedure for finding the equation of an ellipse from its visual graph, which step is required to determine the denominator for the term?
A design engineer is extracting mathematical parameters from a 2D blueprint of an elliptical opening. The blueprint has the opening centered at the origin of a coordinate system. To construct the correct equation in the form , match each visual feature of the ellipse's graph with its algebraic role in the equation.
A technician is using a coordinate system to find the equation of an elliptical part centered at the origin . After identifying the distance from the center to the intercepts on the -axis, the technician must ________ that distance to determine the correct denominator for the term.
When a computer-aided design (CAD) technician is deriving the equation for an elliptical component centered at the origin , the standard procedure for finding the denominator of the term is to simply use the distance from the center to the -intercept without any further calculation.
Learn After
A satellite technician is modeling an elliptical orbit with the equation , where the center is at the origin and the major axis is horizontal. In this standard form, what geometric property of the orbit is represented by the denominator 625?
A mission specialist at a satellite tracking station is modeling an elliptical orbit centered at the origin . In this specific model, the vertices are located at and the sun is positioned at a focus point . Match each orbital parameter with its corresponding value or formula used to determine the standard equation of the orbit.
Orbital Equation Parameters
An astrophysics researcher is documenting the procedure for modeling a planet's elliptical orbit where the sun is at a focus, with a closest distance of 20 AU and a furthest distance of 30 AU. Arrange the following steps in the correct order to derive the standard equation of the ellipse centered at the origin, as demonstrated in the lab manual.
A data analyst at a satellite tracking station is modeling a planet's elliptical orbit centered at the origin. The analyst identifies that the squared semi-major axis is and the squared distance to the focus is . Using the relationship to find the denominator for the term, the analyst determines that the value of is ____.