Satyavati bore Shantanu two sons: Chitrangada and Vichitravirya. The
Harivamsa tells of Bhishma recalling events after Shantanu's death.
During the period of mourning after Shantanu’s death, Ugrayudha
Paurava (usurper of the throne of Panchala) demanded that Bhishma hand
over Satyavati in return for wealth. Bhishma killed Ugrayudha Paurava,
who had lost his powers because he lusted after another's wife.
However, the Mahabharata does not include this event; it only
describes Bhishma crowning Chitrangada as king under Satyavati's
command. Chitrangada was later killed by a namesake gandharva (a
celestial musician).
After Chitrangada's death his young brother Vichitravirya was crowned
king, while Bhishma ruled on his behalf (under Satyavati's command)
until Vichitravirya grew up. Vichitravirya married the princesses of
Kashi-Kosala: Ambika and Ambalika, who were won by Bhishma in a
Swayamvara (marriage choice). The childless Vichitravirya met with an
untimely death from tuberculosis.
An aged Satyavati with her first born Vyasa With no heir to the
throne, Satyavati asked Bhishma to marry the widows of Vichitravirya
(following the practice of niyoga in its narrower sense, as a levirate
marriage) and rule as king. Bhishma refused, reminding Satyavati
of the promise he made to her father and his vow of
bachelorhood. He suggests that a Brahmin could be hired
to father children on the widows, thus preserving the dynasty.
Revealing to Bhishma the tale of her encounter with Parashara,
Satyavati well knew that this was the time to call her son Vyasa to
aid her. Satyavati coaxed Vyasa to have niyoga with his brother's
widows, saying: "from affection for thy brother Vichitravirya,
for the perpetuation of our dynasty, for the sake of this Bhishma's
request and my command, for kindness to all creatures, for the
protection of the people and from the liberality of thy heart, O
sinless one, it behooveth thee to do what I say." After convincing
Vyasa, Satyavati managed ( with great difficulty) to obtain the
consent of her "virtuous" daughters-in-law.[16] In the Mahabharata,
Vyasa agreed immediately to the niyoga. In the Devi Bhagavata Purana
version, Vyasa initially refused Satyavati's proposal. He argued that
Vichitravirya's wives were like his daughters; having niyoga with them
was a heinous sin, through which no good could come. As a master of
"realpolitik", the hungry-for-grandsons Satyavati asserted that to
preserve the dynasty, wrong directives by elders should be followed if
they are going to reduce the sorrow of a mother. Vyasa finally agreed
to that "disgusting task", but suggested that offspring of perversity
cannot be a source of joy.
Birth of Pandu ,Dhritrashtra & vidhur your First point satyavati asked for her grand children not for herself
During the menstrual period of the older queen, Ambika, Satyavati
invited Vyasa to Ambika's bedchamber. However, during niyoga with
Vyasa, Ambika noticed his dark appearance and closed her eyes. Vyasa
declared to Satyavti that due to Ambika's cruelty, her son would be
blind (but strong) and have a hundred sons – later known as Kauravas
(descendants of Kuru). Satyavati considered such an heir to be an
unworthy king, so she asked Vyasa to have niyoga with her other
daughter-in-law. During their niyoga, Ambalika fell pale due to
Vyasa's grim appearance. As the result the child would be wan, Vyasa
told his mother, who begged for another child. In due course, the
blind Dhritarashtra and the pale Pandu were born. Satyavati again
invited Vyasa to Ambika's bedchamber; she remembered Vyasa's grim
appearance (and repulsive odour), and substituted a Shudra (lowest
caste) maid in her place. The maid respected the sage and was not
afraid of him, and Vyasa thus blessed her; her son would be the most
intelligent man, and she would no longer be a slave. Vyasa told
Satyavti of the deception, and then disappeared; Vidura, an
incarnation of the god Dharma, was born to the maid.
Birth of children and grandchildren- Find Here