HLEPP…GIVEN luminosity: 86.814L_s . Temperature:10787K . mass. 3.58 M_s . Radius: 2.69. Period: 130 days and 0.356 years .. orbital radius 0.76841 AU …. what is the planet Radius ___R_E and ____ R_J



Answer :

To find the radius of the planet in units of Earth radii [tex](\(R_E\))[/tex] and Jupiter radii [tex](\(R_J\))[/tex], we need to use the provided orbital radius and the physical characteristics of the star. Here's how we can calculate it:

Given data:

- Orbital radius of the planet [tex](\(r\))[/tex]: \(0.76841\)[/tex] AU

- Stellar radius [tex](\(R_*\))[/tex]: \(2.69\)[/tex] times the radius of the Sun [tex](\(R_\odot\))[/tex]

Step-by-step calculations:

1. Convert the orbital radius to kilometers (km):

- 1 AU (Astronomical Unit) = [tex]\(1.496 \times 10^8\) km[/tex]

- Orbital radius \(r\)[/tex] in kilometers:

[tex]\[r = 0.76841 \times 1.496 \times 10^8 \text{ km}[/tex]

[tex]\[ r \approx 1.149 \times 10^8 \text{ km}\][/tex]

2. Calculate the radius of the star in kilometers:

- Given[tex] \(R_* = 2.69 R_\odot\)[/tex]

- Assume [tex]\(R_\odot = 695,700\)[/tex] km (average radius of the Sun)

- Stellar radius \(R_*\)[/tex] in kilometers:

[tex] \[R_* = 2.69 \times 695,700 \text{ km}\][/tex]

[tex]\[ R_* \approx 1,868,893 \text{ km}\][/tex]

3. Determine the planet's radius in Earth radii [tex](\(R_E\))[/tex]:

- Assume Earth's radius [tex]\(R_E = 6,371\)[/tex] km

- Planet's radius [tex]\(R_{\text{planet}}\) in \(R_E\):\[R_{\text{planet}} = \frac{r}{R_E}\][/tex]

[tex]\[R_{\text{planet}} = \frac{1.149 \times 10^8\text{ km}}{6,371 \text{ km}}\][/tex]

[tex]\[R_{\text{planet}} \approx 18,016\][/tex]

4. Calculate the planet's radius in Jupiter radii [tex](\(R_J\)):[/tex]

- Assume Jupiter's radius [tex]\(R_J = 69,911\)[/tex] km

- Planet's radius [tex]\(R_{\text{planet}}\)[/tex] in \(R_J\)[/tex]:

[tex]\[R_{\text{planet}} = \frac{r}{R_J}\][/tex]

[tex] \[R_{\text{planet}} = \frac{1.149 \times 10^8 \text{ km}}{69,911 \text{ km}}\][/tex]

[tex]R_{\text{planet}} \approx 1,643[/tex]

Summary:

- Planet's radius [tex]\(R_{\text{planet}} \approx 18,016 R_E\)[/tex]

- Planet's radius [tex]\(R_{\text{planet}} \approx 1,643 R_J\)[/tex]

These calculations give us the planet's radius relative to Earth and Jupiter based on the provided orbital radius and the characteristics of the star.