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There are two very common trees referred to as "Satsuma". One is a tangerine, and one is a plum. The two couldn't have more of an opposite care regimen. Since it is not specified which I am answering for, I will answer for each.

Satsuma tangerine fruits take, roughly, 9 months to ripen. A month and a half is not very long in the grand scheme of things. You won't see much change in that short of a time period. Patience, and PLENTY of light will be the key to ripening here.

This article will give you more information on the care of these trees:

https://www.gardeningknowhow.com/edible/fruits/tangerine/mandarin-orange-tree-care.htm

The Satsuma plum will be a very different story. Since there is only one tree, it will never produce fruit. It will only ever produce small, green,hard, round, useless "berries" unless you have a second tree. Once there is a pollinator nearby, you can expect a fruit crop from these.

This article will help with these: https://www.gardeningknowhow.com/edible/fruits/plum/satsuma-plum-care.htm

This can have a few causes. I would recommend applying dolomitic lime, iron sulfate, and a citrus fertilizer at least once per year. Sulfur will help with the flavor quite a bit, as the trees use this to build their flavor profile.

It sounds like there is a fungal infection within the tree. Is the tree beside it the same kind? Do you have photos? It will be hard to judge what is going on without knowing what kinds of treatment that it has, and without seeing the damage.

Please include photos, and just a little more information about the weather, how much water it has received, and about the other trees in the area.

It sounds like a fungal infection, but if the tree right beside it is the same exact tree, then it would be strange not to see signs on that one as well.

For now, this article will help: https://www.gardeningknowhow.com/edible/fruits/plum/plum-trees-not-fruiting.htm

Our experts say in the Spring, fertilize with a 10-10-10 food and then again in the early summer. Simply sprinkle a handful of fertilizer around the base of the plum and water in well. ttps://www.gardeningknowhow.com/edible/fruits/plum/satsuma-plum-care.htm

Several things can cause fruit drop. One occurs when the tree has produced more fruit than it can handle. It will drop some to lighten the load so to speak. Also, stressors such as disease, insects or environmental stresses such as overpruning, drought or excessive rainfall also can cause fruit drop.

https://extension.arizona.edu/sites/extension.arizona.edu/files/pubs/az1492.pdf

https://www.gardeningknowhow.com/edible/fruits/tangerine/mandarin-orange-tree-care.htm

There may be a fertilization problem. Apply a citrus fertilizer in early spring, summer and fall or according to the instructions with the fertilizer.

Deep water adequately and mulch the soil surface to help retain soil moisture and supply soil organic matter.

Here is more:
https://www.gardeningknowhow.com/edible/fruits/citrus/fertilizing-citrus-trees-best-practices-for-citrus-fertilizing.htm