Consider this sentence:
Most historians trace the beginning of the Internet to Sputnik, the first satellite ..........
The correct answer is 2. But why not 1? and why one should use "to" here?
A relative clause identifies or qualifies or describes a preceding noun. It is normally introduced by a relative pronoun or relative adverb. It contains a subject and a properly inflected verb. So the proper way to use a relative clause in your example is Sputnik, the satellite that first orbited the earth. (As was pointed out in a comment, "successfully" is redundant.) The relative pronoun "that" is the subject and "orbited" is the verb.
The second option is to use an appositional phrase with a "to infinitive." Such a phrase is an abbreviated substitute for a particular form of relative clause. In your example, it is abbreviating Sputnik, which was the first satellite to orbit the earth. The subject of the clause is "which" and the verb is "was." If the subject and verb are dropped, we get an appositional phrase with the infinitive retained: Sputnik, the first satellite to orbit the earth.
The first one to do it will win a prize.
The first one to do it wins a prize.
The first one to do it won a prize.
The infinitive in general does not refer to an occurrence of the action in real time; rather it refers to the action abstractly, outside of time. You can see from the examples above that the main verb can be future or past or present. So, the infinitive clause (in bold) is subject-oriented, telling us something about the subject.
Is there a mechanic in town who can repair a 1952 Ford?
--The man to do it is Bob. He knows a lot about old cars.
We can paraphrase this clause with a relative clause:
The man who can do it is Bob.
or with a prepositional phrase:
The man for the job is Bob.
