Given a unit sphere (radius 1), I would like to know how many cones I can pack into this unit sphere. Restrictions: The top of the cone needs to be in the center of origin. The bottom of the cone needs to form a circle on the unit sphere.

I have found a related question, but with a cube: Packing space by cones: Translates best?

I have also tried to find an upper bound myself by performing the following calculation:

Surface of the projection of the base of the cone on the unit sphere:
$$2 \pi r^2(1 + \sin(\theta) \pm \cos(\theta))$$

Surface of the unit sphere: $4 \pi r^2$

Now, a (very high) upper bound would be:

$$\frac{2 \pi r^2(1 + \sin(\theta) \pm \cos(\theta))}{4 \pi r^2}$$

This however does not take into account the restrictions of the shapes, so the actual number will likely be much lower.

Question 1: What would be a closer upper bound

Question 2: If an example value is easier, what would be a realistic number of cones given $\theta = 5^{\circ}$