1. Understanding the problem: We need to find the number of valid stellar classifications for the 9 stars in the constellation Leo, where each star has a distinct classification from its connected stars. Then, we need to compute the number of positive integer divisors of this number.

2. Analyzing the pentagon subset: Consider the 5-star pentagon subset of the constellation, denoted as $ABCDE$. We need to classify these stars such that no two connected stars have the same classification.

3. **Classifying $A$ and $B$**: There are 7 possible classifications for $A$ and 6 remaining classifications for $B$, giving us $7 \times 6 = 42$ ways to classify $A$ and $B$.

4. **Classifying $D$:
- Case 1**: $D$ has the same classification as $A$ or $B$. There are 2 choices for $D$ (same as $A$ or $B$), 6 choices for $C$ (different from $A$ and $B$), and 5 choices for $E$ (different from $A$, $B$, and $C$). This gives $2 \times 6 \times 5 = 60$ ways.
- Case 2: $D$ has a different classification from $A$ and $B$. There are 5 choices for $D$, 5 choices for $C$ (different from $A$, $B$, and $D$), and 5 choices for $E$ (different from $A$, $B$, $C$, and $D$). This gives $5 \times 5 \times 5 = 125$ ways.

5. Total classifications for the pentagon: Summing the two cases, we get $60 + 125 = 185$ ways to classify the pentagon given the classifications of $A$ and $B$. Therefore, the total number of valid classifications for the pentagon is $42 \times 185 = 7770$.

6. Classifying the remaining 4 stars: Each of the remaining 4 stars can be classified in 6 ways (since they are connected to at most 3 other stars, and there are 7 classifications in total). Thus, the total number of valid classifications for the entire constellation is:
$$7770 \times 6^4 = 7770 \times 1296 = 10077120$$

7. Finding the number of positive integer divisors: To find the number of positive integer divisors of $10077120$, we need its prime factorization:
$$10077120 = 2^7 \times 3^4 \times 5^1 \times 7^1$$
The number of positive integer divisors is given by multiplying the incremented exponents:
$$(7+1)(4+1)(1+1)(1+1) = 8 \times 5 \times 2 \times 2 = 160$$

The final answer is $\boxed{160}$