To find the eccentricity of the orbit, we need to use the given conditions and the basic equations of orbital mechanics.

Given:

• $\frac{r_0 \cdot v_0^2}{\mu} = 3$
• $\beta_0 = 10^0$ (which means $\beta_0 = 1^\circ$)
• Perigee radius $r_p = 9000 \, \text{km}$
• Earth's radius $R = 6400 \, \text{km}$
• Standard gravitational parameter $\mu = 398601 \, \text{km}^3/\text{s}^2$

Key Equations:

1. Orbital energy equation: $\epsilon = \frac{v_0^2}{2} - \frac{\mu}{r_0}$

2. Eccentricity equation: $e = \frac{r_a - r_p}{r_a + r_p}$ where $r_a$ and $r_p$ are the apoapsis and periapsis distances, respectively.

3. Equation relating $r_0$, $v_0$, and $\mu$: $r_0 \cdot v_0^2 = 3\mu$

Step-by-Step Solution:

1. From the given equation: $r_0 \cdot v_0^2 = 3\mu$ $v_0^2 = \frac{3\mu}{r_0}$

2. The flight path angle $\beta_0$ relates the tangential and radial velocities: $\tan(\beta_0) = \frac{r_0 \dot{\theta}}{\dot{r}}$ where $\dot{\theta} = \frac{v_0}{r_0}$.

3. For small angles, $\tan(\beta_0) \approx \beta_0$, so: $\beta_0 = \frac{r_0 \dot{\theta}}{\dot{r}}$

   Given that $\beta_0 = 1^\circ$, we can approximate:

   $r_0 \approx \frac{r_p (1 + e)}{1 - e}$

4. We can calculate the value of eccentricity using the relation between the perigee, apogee, and the semi-major axis.

Let's calculate the eccentricity numerically.The solution for eccentricity $e$ based on the given relation is:

$e = \frac{r_0 - 9000}{r_0 + 9000}$

Next, we need to find $r_0$ using the provided condition $r_0 \cdot v_0^2 = 3\mu$. Since $v_0^2 = \frac{3\mu}{r_0}$, we can substitute this into the equation for eccentricity.

Let’s compute $r_0$ using the known parameters and then calculate the corresponding eccentricity value.It seems that the assumption $r_0 = r_p = 9000$ km results in an eccentricity of $e = 0$, which indicates a circular orbit. However, given the context and the provided options, this outcome doesn't match any of the choices provided.

Let's take into account that the orbital speed $v_0$ isn't directly tied to $r_p$, and re-calculate the eccentricity assuming a different $r_0$ corresponding to a non-circular orbit.

I'll revise the calculations to find the correct eccentricity.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, if you'd like, I can walk you through the steps to manually calculate the eccentricity or address any other questions you might have. Let me know how you'd like to proceed!

Here are some related questions you might be interested in:

1. How is the orbital speed of a satellite calculated?
2. What are the different types of orbits based on eccentricity?
3. How does the perigee and apogee affect the orbit shape?
4. What is the significance of the flight path angle in an orbit?
5. How can you derive the orbital energy equation?

Tip: Understanding the relationship between orbital parameters like eccentricity, speed, and radius is crucial for analyzing satellite orbits.