In the Hindu planetary system, Rahu and Ketu take an invisible place. But we use to find their periodic presence in houses. In modern science, asteroid belts are between Mars and Jupiter. Asteroids are basically the ones which failed to form as a planet or parts of destroyed planet. Rahu and Ketu also destroyed individual in the shape of snake. Asteroid belt is also a ring around the sun with dust and stones. Is this asteroid belt considered as invisible Rahu and Ketu in our ancient Hinduism?
No, Rahu and Ketu are not related to the asteroid belt. Rather, they denote the two lunar nodes, the points where the Moon's orbit around the Earth crosses the ecliptic, the path of the Sun in the sky. Here is what this book says:
[T]he orbit of the Moon intersects with the ecliptic at two points[.]... These two points are referred to as the ascending and descending nodes of the Moon. They are called Rahu and Ketu.
Now when the Sun or Moon passes through one of the lunar nodes, an eclipse occurs. This is Rahu and Ketu's revenge on the gods Surya and Chandra, as I discuss in my answer here.
Rāhu-Ketu meant different things in different Hindu scriptures as noted in A Concise History of Science in India. It's only from the time of Indian astronomer/astrologer, Varāhamihira, they came to assume the meaning: the ascending-descending nodes where the Moon crosses the ecliptic.
2. Astronomy (S. N. Sen)
Astronomical Knowledge as Revealed in the Saṃhitās, Brāhmaṇas and Sūtras
Rāhu and Ketu
To the seven planets, including the sun and the moon, the Indians added Rāhu and Ketu, to formulate their Rāhu-Ketu theories of eclipses. The word Rāhu, in the sense of a planet, appears in the Atharvaveda and the Chāndogyopaniṣad apparently with no astronomical meaning, but in the Yājñavalkyasmṛti it does so in the astronomical sense. The word Ketu also appears in the Atharvaveda in the sense of any unusual or striking phenomenon such as comet, meteor or a falling star. Astronomical saṃhitās, however, do not mention Ketu as a cause of eclipse. Rāhu and Ketu, along with planets, are mentioned in the Mahābhārata, though Ketu is omitted in the Rāmāyaṇa. From Varāhamihira onwards, in true astronomical circles, Rāhu and Ketu meant the ascending and the descending node respectively of the moon.
The Sizes of The Sun and The Moon, Their Distances from The Earth, Eclipses and Parallax
We have seen that from Vedic times the Indians were interested in the phenomena of solar eclipses. Good descriptions of them have been preserved in the Brāhmaṇa literature, but scientific studies of them were perfected in the siddhānta period. The mythological and the Purāṇic Rāhu-Ketu theory which had become firmly entrenched in the minds of laymen and even astronomers and continued long thereafter gave place to the scientific one based on the relative positions of the sun, the moon and the earth and the shadows cast by the moon on the earth or by the earth upon the moon. Contrary to the Purāṇic idea of Rāhu devouring the sun or the moon, Āryabhaṭa says: 'The moon obscures the sun and the great shadow of the earth obscures the moon. When at the end of the true lunar month (at new moon) the moon, being near the node, enters the sun, or when, at the end of half month (at full moon), the moon enters the shadow of the earth, that is the middle of the eclipse, which occurs sometimes before and sometimes after the exact end of the lunar month or half month.' Varāhamihira explodes the Rāhu-Ketu myth and says that the real cause of a lunar eclipse is the entry of the moon into the earth's shadow and likewise at the solar eclipse the moon enters the sun.
A very clear and lucid exposition of eclipses is given by Bhāskara II in his Siddhānta-śiromaṇi, section Golādhyāya, chapter Grahavāsanā. He says: 'The orbit of the moon is below that of the sun. Just as a cloud moving from behind covers the sun so does the moon moving faster covers the sun from behind. That is why the western part of the sun is eclipsed by the moon first and the eastern side released last. Owing to differences in the latitude eclipses are sometimes seen and sometimes missed.' Then follows a discussion on the need of parallax correction and so on.
Fig. 1.7. Eclipses in the Sun-Earth-Moon planetary system
Hinduism worked out the most detailed and useful mythology of eclipses, which greatly influenced not only Eastern but also European astrology and astronomy, by introduction of lunar node concept. Hindu astrology considers the planets and some other cosmic phenomena as Graha – cosmic "influencers" on the history and all living beings. The Navagraha ("nine influencers") includes the Sun, the Moon, which are considered as planets, Mars, Mercury, Jupiter, Venus, Saturn, as well as positions in the sky – Rahu (north or ascending lunar node) and Ketu (south or descending lunar node).
Fig. 1.21. Rahu and Ketu – the demons of ascending and descending lunar nodes
Rahu, the demon of ascending lunar node is the head of the Demon Snake that swallows the Sun or the Moon causing eclipses. According to legend the asura Rahu in the form of giant Demon Snake drank some of the divine nectar Amrita. But before the nectar could pass his throat, Mohini (the female avatar of god Vishnu) cut off his head. The head, however, remained immortal and is called Rahu, while the rest of the Demon Snake’s body became Ketu – the demon of descending lunar node. It is believed that Rahu’s immortal head occasionally swallows the Sun or the Moon causing eclipses and then the Sun or Moon passes through the opening at the neck, ending the eclipse. It is believed that Rahu and Ketu had a tremendous impact on the Earth’s creation and still seriously influence human lives and fates.