Science of gotra

Kula and Gotra, like clans and tribes are methods of identifying  kinship among people it can be called as the social arrangement or the social science.

In all Community Generally the child gets the gotra from his father
The gotras of brahmins can be traced back to the saptharishis
And for other communities they have their respective clans

For example
There is an another system of arrangement in costal karnataka region which is called as " bali " In which the child inherits the gotra from his/her mother.

There are three sources of Gotras.

Your family clan learned from the elders in the family.

When a woman and man marries, the woman joins the Gotra of her husband.(This doesn't go in case of " bali " The woman still inherits the bali of her mother only and her children gets the same bali of herself)

If you don’t have a Gotra and are initiated by a guru, then you take the guru’s Gotra since through the process of initiation (dīkṣa) you become the spiritual child of the guru.

You take your astrological Gotra based on your nakṣatra or birth asterism. Check your birth nakṣatra on the and find your Rishi and Gotra

1. Marichi (light)
Ashwini, pushya, swati, Abhijit (Not always used)

2.Vashishtha(possessor of wealth)
Bharani, ashlesha, vishaka, shravana

3.Angiras (The fiery one)
Krittika, magha , Anuradha, Dhanistha

4.Atri (one who devours)
Rohini, purva phalguni, dhanistha, shatabhisha

5. Pulasthya (one who has smooth hair)
Mrigashira, uttara phalguni, moola, poorva bhadrapada.

6. Pulahu (connector of space)
Adra, hasta, purva ashadha, uttara bhadrapada.

7.Kratu (the inspirer)
Punarvasu, chitra, uttara ashadha, revati

A science behind it:

different studies revealed that children born from parents who were closely related increases the risks of having a baby with potentially life-threatening birth defects.

Marrying within a family can lead to abnormalities in your future offspring. This is because within a family, certain genetic traits remain dormant and are known as recessive genes (they are not seen as a disease or condition).

When two people having the same genes marry and reproduce, these once dormant recessive (ineffective) genes double, making them more likely to cause genetic abnormalities in children.

Proof
Austrian hamsburg dynasty was so inbred( marriages inside relation) that  they had an extremely disabled king called king Charles 2nd of spain

Smriti chandrika
" If one does not know or forget's one's gotra dharma shastra recommends the adoption of gotra of great rishis kashyapa or jamadagni. "

So during ceremonies if you don't have your particular gotra then the priest would allot you to kashyapa gotra or gotra of jamadagni.