The orbit of a certain planet is given by the ellipse (x-h)^2/a^2 + y^2/b^2= 1 , where a= 1.15, b= 4.63, and h= 2.96. What is the longest distance (Aphelion) that this planet reaches with respect to its star as it orbits around it?

To determine the longest distance (Aphelion) that the planet reaches with respect to its star, we can use the information provided about the elliptical orbit.

The equation of the ellipse is given by:

\[
\frac{(x - h)^2}{a^2} + \frac{y^2}{b^2} = 1
\]

Here, \( h = 2.96 \), \( a = 1.15 \), and \( b = 4.63 \).

### Key points:
- The **Aphelion** is the point in the orbit where the planet is farthest from the focus of the ellipse (which typically corresponds to the star in this context).
- The distance from the center of the ellipse to the Aphelion is given by \( h + a \). This is because at the Aphelion, the \( x \)-coordinate is at its maximum value \( x = h + a \), where the \( y \)-coordinate is 0.

### Calculation:

The longest distance (Aphelion) is therefore:

\[
\text{Aphelion} = h + a = 2.96 + 1.15 = 4.11 \text{ units}
\]

So, the longest distance (Aphelion) that the planet reaches with respect to its star is **4.11 units**.

Would you like more details or have any questions? Here are 5 related questions to further explore the topic:

1. How is the shortest distance (Perihelion) of the planet from the star calculated?
2. What is the eccentricity of this ellipse, and how does it relate to the orbit's shape?
3. How does the value of \( b \) affect the orbit of the planet?
4. What is the significance of the parameter \( h \) in the ellipse equation?
5. Can this method be applied to calculate the Aphelion of orbits that are not elliptical?

**Tip:** The semi-major axis \( a \) of an ellipse is always half of the longest diameter, and it plays a crucial role in determining the properties of the orbit.

Explore how to calculate the Aphelion distance of a planet orbiting in an elliptical path defined by (x-h)^2/a^2 + y^2/b^2 = 1. Understand the equation's parameters a, b, and h, and learn the step-by-step process to determine the farthest point from the star. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation