There are varying descriptions in scripture about the appearance of Srimati Radharani on the earth; one reason for this is that the appearances took place in different kalpas.
In his Sri Lalitä Mädhava, Srila Rüpa Gosvämi relates the appearance of Srimati Rädhärani as explained by Pürnamäsi Devi to Gärgi, the daughter of Garga Muni. The details of this appearance were known to Pürnamäsi Devi, as she arranges all the pastimes of the Lord, and she shared this information only with Mother Yasodä and Rohini Devi.
Vindhya-parvata* was envious of the great Himälaya Mountains, because Himälaya, having Pärvati as his daughter, had the opportunity to get Lord Mahädeva as his son inlaw. Vindhya-parvata desired to have a fortunate daughter whose husband could defeat Mahädeva in fighting, and who would thus attain the position of Räjendra (Emperor). To fulfil this desire, Vindhya-parvata performed severe austerities in order to please Lord Brahmä. After some time Lord Brahmä appeared before him and asked him what kind of boon he sought. Vindhya replied that he wanted to have a daughter who carried the blessing that whomever she married could defeat Mahädeva in battle. Because Brahmä was very pleased with him, immediately he said, “Tathästu: so be it.”
Vindhya was very happy to receive the boon from Lord Brahmä. But after offering the boon Lord Brahmä was thinking, “Who is that person who can defeat Mahädeva in battle? It is not possible…” He was worried now about the boon he had granted. Then he realized that Lord Krishna’s pastime was going to take place on the earth: only He can defeat Mahädeva in battle. Brahmä thought, “Lord Krishna’s eternal consort is Srimati Radharani. If Vindhya-parvata can get Rädhäräni as his daughter, only then can my boon be justified. But how can this happen? Srimati Kirtidä is the eternal mother of Rädhärani. How can Vindhya then get Her as his daughter?”
Worrying about how his boon would come to fruition, Lord Brahmä underwent severe austerities to please Srimati Rädhäräni. When She was pleased with him, he requested Her to please appear as the daughter of Vindhya-parvata. Rädhärani agreed, and Yogamäyä Devi arranged for two girls—Rädhäräni and Candrävali, who were were already in the wombs of the wives of Vrsabhänu and Candrabhänu—and put them in the womb of the wife of Vindhya-parvata.
As a result, the wife of Vindhya begot two daughters who were very beautiful.In the meantime Lord Krishna had appeared in Mathurä. By the order of the Lord, Väsudeva took Krishna to Gokula and left Him with Yasodä, who had given birth to a daughter. Väsudeva exchanged Krishna with the daughter of Yasodä, and took her to the prison cell where he and Devaki were being held by Kamsa.
Hearing that Devaki’s eighth child had taken birth, immediately Kamsa came to the spot and snatched the girl from the lap of Devaki. When Kamsa was ready to throw the girl on the stone and thus smash her, the girl escaped from his hand and appeared in her form as Yogamäyä Devi.
She declared loudly, “Kamsa! You were the demon Kälanemi in your previous birth. That Lord who had killed you with His disc in the battle, who is known by the learned personalities as the most worshipable of the demigods, and who is cause of the whole cosmic manifestation, today He has appeared on the earth to give pleasure to all the living entities.
Kamsa! You should know one more thing: by tomorrow, eight mähä-sakti—greater and more attractive than me—will also take birth on the earth. Their names are Rädhä, Candrävali, Lalitä, Visäkhä, Padmä, Saibyä, Syämalä and Bhadrä. Among these eight mähä-sakti, two will be most famous for their great qualities. Whoever marries these two girls will be most fortunate: he will be Räjendra, and will be so powerful that he will defeat even Mahädeva in battle.”
After saying this, Devi disappeared from there and left for Vindhya Mountain. When Kamsa heard this from Devé he ordered Putana to kill all newborn males and abduct all newborn females. According to the order of King Kamsa, when Pütanä was searching for such children she found two most beautiful baby girls.
After Vindhya-parvata’s wife gave birth to two daughters, Vindhya arranged a samsakara ceremony for these two girls. A brähmana was performed the yajna with the two daughters present. When Pütanä saw these two most beautiful girls, immediately she picked them up and started to fly away in the sky. King Vindhya became very worried and ordered the brähmanas to chant mantras to kill the demon who was stealing his daughters. By the order of Vindhya, the brähmanas chanted the mantra, which caused Pütanä to gradually become weaker and weaker. She was not able to hold both the baby girls, and threw one of them into a river below, which flowed in the kingdom known as Vidarbha. Bhismaka, the king of Vidarbha, found this girl and kept her with him until the age of five.
At that time Jämbavän was residing in both Vindhya and Govardhana Mountains. By the order of Vindhya, Jämbavan went to Vidarbha and took this girl back. She was known as Candrävali.
While Putana was still flying with the other girl in her arms, her weakness increased, and upon reaching Vraja she fell to the ground. At that time Pürnamäsi Devi recovered that baby girl from Pütanä and gave her to Mukharä, telling her, “She is the daughter of your son-in-law, Vrsabhänu, so you should take care of Her.”
Thus from that day She became known as Rädhä, the daughter of King Vrsabhänu. Pürnamäsi had recovered five more girls from Pütanä. They were Lalitä, Padmä, Bhadrä, Saibyä, and Syämä.
Thus Srimati Rädhäräni and Candrävali appeared along with their sakhis. This information was given by Närada Muni to Pürnamäsi Devi.
There is another description of the appearance of Srimati Rädhäräni on this planet.
Once Sürya Deva (Bhänu) worshipped Lord Sré Hari with a desire to get Golokesvari as his daughter. The Lord was pleased with Sürya and agreed to fulfill his desire. On the earthly planet, in Gokula Mähävana, there is a village called Ravel. In that village there was a king amongst the cowherds named Mähäbhanu. He had five sons: Candrabhänu, Ratnabhänu, Vrsabhänu, Subhänu and Pratibhänu. Each of them was very sober, self-controlled, and were great Vaiñëavas. Among these five brothers, Vrsabhänu was selected to take up the position on the throne. This Vrsabhänu was non-different than Sürya Deva (Bhänu Deva). King Vrsabhänu performed hundreds of Räjäsuya yajnas for the pleasure of Lord Sri Hari, and observing religious practices in all respects. He was a great räjärsi, and everyone worshipped him for his great qualities.
At that time there was another very rich cowherd man living in Vraja called Bindu. His wife’s name was Mukharä. Bindu had fathered five sons and three daughters from the womb of his wife Mukharä. The five sons were Bhadra-kirti, Candra-kirti, Mähäbala, Mähä-kirti and Sridäma. His daughters’ names were Bhänu-mudrä, Kirtimati and Kirtidä. According to some Puräëas, another name of Kirtidä is Kalävati.
King Vrsabhänu married Kirtidä, the daughter of Bindu. Even after many years had passed still they had no child. Both of them were passing their days in great distress and trying their best to conceive a child by performing different kinds of yajnas, giving in charity, peforming special püjas and arcana, and traveling to different holy places. But still they were not successful.
Finally the most chaste Kirtidä advised her husband to worship Devi Katyäyäni. As per her advice, King Vrsabhänu commenced worship of Katyäyäni Devi at the foot of Govardhana on the bank of the River Yamunä. He performed austerities by following mauna-vrata and abstaining from food for long periods of time.
One day while absorbed in meditation during his vrata, King Vrsabhänu heard a sound vibration:
harinäma vinä vatsa
karna-suddhir na jäyate
tasmät sreyaskaraà räjan
hari-nämänukirtanam
grhäna hari-nämäni
yathä-kramam vinindita
“O Vatsa (my dear son)! With out accepting harinäma the ears do not get purified. To chant the holy name of Hari is most auspicious. Accept the holy name of Hari properly.”
Then king Vrsabhänu went to the äsrama of Kratu Muni, which was situated on the bank of Yamunä, and accepted harinäma from Kratu Muni, who explained so many different rules and regulations. The mantra given by Kratu Muni was:
Hare Krishna Hare Krishna Krishna Krishna Hare Hare
Hare Räma Hare Räma Räma Räma Hare Hare
Then Kratu Muni told the king,
iti sodasaka-nämätmaka-nämnäm
tri-käla-kalmasäpaham
nätah parataropäyah
sarva-vedesu vidyate
(Brahmandò Puräna, Uttara-khanda, Rädhä-hrdaya)
After giving the mähä-mantra to the king, Kratu Muni told Vrsabhänu,
säkto vä vaisnavo väpi
sauryo vä saiva eva vä
gänapatyo labhet karna-
suddhim nämänukirtanät
Whether Sakta or Saiva, Saurya or Vaisnava or Gänapatya, in any case only harinäma kirtäna brings purification to the ears. Unless one takes harinäma at the beginning and purifies the ears, no other mantra will give any result. In whose earhole the holy name of Lord Hari has not entered, his ears are contaminated like low-born sabaras. O king! The harinäma which I have offered to you has been glorified by Srila Vyäsadeva when once he was asked by Süta Gosvämi. Brahmä has also taught the process of chanting harinäma to Angirä Muni. You should engage yourself in chanting this mantra with full concentration.”
After receiving this mähä-mantra from Kratu Muni, Vrsabhänu offered his prostrated obeisances and came back to the bank of the Yamunä where he sat and chanted the mantra with devotion.
After some time Katyäyäni Devi, being very pleased with King Vrsabhänu, appeared in front of him. The king offered his obeisances and prayers unto Katyäyäni. When Devi asked the king what boon he sought, he replied, “Simply by having your darsana, all my desires are fulfilled. Still, if you are pleased to give me some boon then what can I say? The desires of my heart are known to you!”
After hearing the words of Mahäräja Vrsabhänu, Katyäyäni offered the king an oval box in the shape of a golden egg which was radiant like thousands of suns. Katyäyäni Devi then disappeared.Mahäräja Vrsabhänu returned to his home with great jubilation.
After some time, as the result of the service to harinäma—which was the essence of the vrata of Kirtidä and Vrsabhänu, Srimati Rädhäräni appeared on the most auspicious day at the most auspicious moment. She was very tender and most beautiful like a golden campaka flower.
The third description of the appearance of Srimati Rädhäräni is told as follows: Once, King Vrsabhänu was immersed in the waters of the Yamunä, meditating upon the Lord. Suddenly a one-thousand-petalled lotus flower came floating towards him and touched his body. When he opened his eyes, he saw the most beautiful molten-goldcolored girl lying on the lotus and moving Her hands and legs. As the king had no child he was very happy to receive this girl. He brought Her to his palace and gave Her to his wife Kirtidä. Thus Srimati Rädhäräni appeared in the earthly home of Her eternal parents, Kirtidä Sundari and Mahäräja Vrsabhänu.
https://www.youtube.com/results?search_query=Radha+Appearance
https://iskcondesiretree.com/profiles/blogs/why-srimati-radharani-descended-from-goloka
https://www.slideshare.net/search/slideshow?searchfrom=header&q=Radha+Appearance
