How did Voyager catch up to the Equinox?

Question

In the Equinox episodes of Star Trek: Voyager the rationale for sacrificing the eel-like aliens was in order to attain higher warp speed and thus arrive to the Alpha quadrant faster, but in the finale the Voyager crew manages to catch up to the Equinox. How was Voyager able to catch up? If the Equinox ends up not being faster than Voyager even after sacrificing the eel-like aliens, what is the rationale for the Equinox crew to abandon Voyager and strike out on its own?

Answer 1

The first part of lucasbachmann's answer covers the main point, but I thought I'd add a bit more detail.

Toward the end of the first episode of the two-part story, Janeway ordered Seven of Nine to go to the USS Equinox's engine room and take its warp core modifications offline.

JANEWAY: (to Seven) Go to their engine room. Take those warp core modifications offline.

_{Star Trek: Voyager - S05E26 - "Equinox"}

Seven carried out this order, so when the Equinox made its escape, it only did so at regular warp speeds.

CHAKOTAY: Chakotay to Seven of Nine. What's your status?

SEVEN: I've dismantled the antimatter injectors, but I'll need several minutes to neutralise the dilithium matrix.

CHAKOTAY: We don't have much time.

SEVEN: Understood.

_{Star Trek: Voyager - S05E26 - "Equinox"}

When Ransom's crew reinstated the warp core modifications in the second episode, they learned that they still couldn't get them to work, because Seven had also encoded the power relays to keep them offline. As a result, the Equinox was limited to regular warp speeds throughout that episode.

RANSOM: Infuse the enhanced warp drive with twenty isograms of the compound.

LESSING: Aye, sir. Warp drive ready.

RANSOM: Engage.

(Shudder then a jolt.)

RANSOM: What happened?

GILMORE: The power relays are offline. They've been encoded.

_{Star Trek: Voyager - S06E01 - "Equinox, Part II"}

For reference, in the Voyager pilot, the USS Voyager was stated to be an Intrepid-class vessel with a sustainable cruise velocity of warp 9.975.

STADI: That's our ship. That's Voyager. Intrepid class. Sustainable cruise velocity of warp factor nine point nine seven five.

_{Star Trek: Voyager - S01E01 - "Caretaker"}

Whereas, according to this article from StarTrek.com, the USS Equinox was a Nova-class vessel with a top speed of warp 8.

The U.S.S. Equinox NCC-72381 was a top-of-the-line Nova-class science vessel. The Nova-class replaced the aging Oberth-class starship used for the past century. At a length of around 222 meters long, 8 decks, and a crew of 78, the ship was relatively small compared to other ships of the time.

It had a top speed of warp 8, which was efficient for general scientific missions. Two shuttlecraft and an optional “wave rider” shuttle, located on the bottom of the “saucer” section, made for a decent auxiliary craft complement. Armaments included 11 Type X phaser arrays and 3 photon torpedo launchers, giving the ship the ability to defend itself decently during its voyages.

_{Inside the U.S.S. Equinox NCC-72381}

In fact, at one point, Janeway stated that without its enhanced drive, the Equinox was only capable of warp 6. It's unclear if this was a mistake by the writers, or if the Equinox's top speed was reduced by the damage the ship had taken. Either way, it was significantly slower than Voyager, which explains why Voyager repeatedly caught up to it.

JANEWAY: Our first priority is to find Ransom. Still no sign of nucleogenic particles?

CHAKOTAY: Not yet.

JANEWAY: Then he couldn't have gotten far. Without his enhanced drive, his ship's only capable of warp six.

_{Star Trek: Voyager - S06E01 - "Equinox, Part II"}

With it's enhanced drive fully operational, the Equinox was apparently capable of travelling over 10,000 light years in under two weeks.

RANSOM: We examined the remains and discovered it could be converted to enhance our propulsion systems. It was already dead. What would you have done? We travelled over ten thousand light years in less than two weeks. We'd found our salvation. How could we ignore it?

_{Star Trek: Voyager - S05E26 - "Equinox"}

By comparison, Janeway stated that even at maximum speeds, it'd take the Voyager 75 years to travel the roughly 70,000-light-year distance between the Caretaker's array and Federation space. Doing a very rough calculation, that'd make the Equinox somewhere in the region of 250 times faster than Voyager, when utilising its enhanced warp drive.

KIM: Captain, if these sensors are working, we're over seventy thousand light years from where we were. We're on the other side of the galaxy.

[...]

JANEWAY: We're alone in an uncharted part of the galaxy. We have already made some friends here, and some enemies. We have no idea of the dangers we're going to face, but one thing is clear. Both crews are going to have to work together if we're to survive. That's why Commander Chakotay and I have agreed that this should be one crew. A Starfleet crew. And as the only Starfleet vessel assigned to the Delta Quadrant, we'll continue to follow our directive to seek out new worlds and explore space. But our primary goal is clear. Even at maximum speeds, it would take seventy five years to reach the Federation, but I'm not willing to settle for that. There's another entity like the Caretaker out there somewhere who has the ability to get us there a lot faster. We'll be looking for her, and we'll be looking for wormholes, spatial rifts, or new technologies to help us. Somewhere along this journey, we'll find a way back. Mister Paris, set a course for home.

_{Star Trek: Voyager - S01E01 - "Caretaker"}

Answer 2

Edit: I misread the question. In the scope of the chase within the episode the answer is the Equinox is a very damaged ship using regular warp.

Transcript from http://www.chakoteya.net/Voyager/525.htm

KIM: Casualty reports are coming in. Two dead, thirteen wounded. We took heavy damage to the engines.

JANEWAY: The Equinox?

KIM: They've gone to warp.

JANEWAY: Any sign of nucleogenic particles?

KIM: No.

JANEWAY: Then they haven't engaged their enhanced warp drive yet. Keep looking for (The sound of new fissures opening.)

---

While it is not the same question so I'm not going to consider it a duplicate - Voyager's large distance jumps are well documented. See Did Voyager "shave off" more years than were necessary?

Note "Equinox" comes before "Dragon's Teeth" and "The Voyager Conspiracy".

The Equinox crew knew what they were doing was unethical but it was also consistently faster. Voyager just gets lucky hits for large jumps. (Of course so did Equinox in its own way.)

Traveling with Voyager would be a loss of independence and legal consequences. Though I suspect a good lawyer could get them acquitted for not knowing those creatures were intelligent - which I recall Voyager figured out. The Federation no doubt has stronger animal rights than we do - but people still need to eat.

Odds are there was some other shady stuff done on their trip too.