What shastra/Scripture describe the astronomical calculations and the procedural details of Upakarma.
Upakarma or Shravani उपकर्म / श्रावणी) is a Vedic ritual celebrated every year generally on the full Moon day of the Lunar Month Sravana (श्रावण) that falls during August–September months of English calendar which also coincides with the monsoon season. Upakarma is referred to as Avani Avittam in Tamil, Jandhyala Punnama in Telugu, and Janivarada Habba in Kannada.
Upakarma is a Sanskrit word (Upa + Karma). Literally Upa means before and Karma means an action. Upakarma means an action performed before beginning the Vedic studies. Upa also refers to nearness or close touch with Upanishads (Vedas). In simple terms Upakarma refers to changing of sacred (Holy) thread called Yagnopaveetham on this day by the three varnas (Brahmins, Kshatriyas and Vysyas). See Here
Upakarma Vidhi (Procedure) -
संप्राप्ते श्रवणस्यांतेय पौर्णिमास्याम दिनोदये , स्नानं कुर्वन्ति ।
मतिमान स्मृति श्रुति विधानतः / उपाकर्मादिकं प्रोक्तंऋषिनाम चैव तर्पणं।।
शुद्राणाम मंत्र राहितं स्नानं दानं च श्याशतै उपकर्माणि कर्तव्यामृशीणं चैव पूजनं।।
Meaning - On the arrival of shravan Month Pournami’s Sun Rise, Wismen perform Upaakarmaadika Karmaas including Tarpanam as per the Intructions ofVeda Shastraas. After the Upakarma, the duty is to perform Puja.
To Rigvedis, ‘Upaakarma Kaala’ is determined on Shravana Shukla Paksha Purnima in Shravana Nakshatra and Panchami’s Hasta Nakshatra;
To Yajurvedis also, Upaakarma Kaala is suitable to Shravana Purnami’s Shravana Nakshatra.
As regards Saama Vedis, the decision is as follows:
सामवेदिनाम भाद्रपद शुक्ले हस्त नक्षत्रं मुख्यकालः।
संक्रांत्यादि दोषांणं तत्र संभवे श्रावण अस्ये हस्तोग्राह्य इति ।।
Nirnaysindhu निर्णयसिन्धु / Dharma Sindhu धर्म सिन्धु
Saam Vedinaam Bhadrapada Shukley Hastaa Nakshatram Mukhyakaalah, Sankraantyaadi doshana tatraa sambhavey Shraavana aasey Hastograahya iti Nirnaya Sindhuh/
Meaning - To Saama Vedikaas, Bhadra pada Shukla Paksha Hastaa Nakshatra is the best option for performing Upaakarma;
To Atharva Vedis, both the options of Shraavana Purnima and Bhaadrapada Purnima are equally acceptable. Purnima however has to be Udaya Vyapini or should span upto Sun Rise next.
Upaakarma Saamaanya Nirnaya or General Rule-
The normal practice of performing Upaakarma to ‘Sarva Shaakhaas’ is in Shravana- Bhadrapada Maasaas. See Full Details
Rigveda Upakarma: Should be done on the day in shraavana where shravana is the star and spans at least from sunrise to mid day.But not in Uttara Ashadha if present during sunrise.
श्रवणे श्रवणायाम् उपाकर्म कुर्वीत आश्वलायन गृह्य सूत्र
Samaveda Upakarma: Should be done on hasta nakshatra on bhadrapada shuklapaksha
प्रौष्ठपदी हस्तायाम् वा श्रवणे श्रवणायां उपक्रमण= गोभिल गृह्य सूत्र
Krishna Yajurveda: i.e. Taittiriyas & Maitrayanis should do on shravistha, dhanistha,(avittam in Tamil),on shraavana month,whether it falls on purnima yukta Pratipada (Pratipat)during sunrise (Telugu- paadyami) or prathama.
Shukla Yajurveda: i.e. Vajasaneyi Kanvas and Madhyandinas do in purnima day if purnima is present up to evening.But if punima and pratipada both are present on next day(para viddha) they should do on previous day when chaturdashi stays in day and then purnima comes after mid day(purva viddha).Under no condition they should do/avoid doing in krishna paksha pratipat or broadly speaking they should finish upakarma before approach of krishna paksha.but they postpone upakarma to the next day if purnima is having Uttara Ashadha nakshatra.
श्रवणे पौर्णमास्यां उपाकर्म =पारस्कर गृह्य सूत्र
Atharva Veda: They do on the day when purnima lasts during sunrise.(udaya vyapini tithi)
If there occurs Eclipse and sankranti when sun enters Leo (masa pirrapu/avani 1st in Tamil) during purnima,the people having upakrama for that day should be finished prior to punima when hasta nakshatra is available(generally shravana shukla Panchami/shashti.and for samaveda,if there is sankramana(purattasi 1st) to kanya in bhadrapada then shraavana month's hasta should be considered.
If shraavana suddha Panchami(hasta) day is having sankramana/sankranti and purnima is having eclipse then upakarma is postponed to Samaveda upakarma day,i'e. bhadrapada hasta nakshatra but only Vajasaneyi Shukla Yajurveda people can do upakarma on the Panchami(hasta) having simha sankramana and avoid upakarma on purnima having eclipse.
The details of upAkarma initially is taken from grhya sutras but as they are short aphorisms, the procedures is given in elaborate in the ritualistic books. Hence respective grhya sutra is to be referred for upAkarma and related rituals. The whole procedure of upAkarma is very long hence it is not possible to include in the post.
For date calculations:
A detailed procedure could be found in the books like
upAkarma paddhati (for respective veda)
aahnika krama (for respective veda)