The aircraft took off from Muscat at 6.35am (2.35am GMT or 8.05am IST) for what is expected to be a 16-hour, 1,465 kilometre journey over the Arabian Sea to Ahmedabad in India.
Pilot Bertrand Piccard was at the controls, taking over from fellow Swiss aviator Andre Borschberg.
On Monday Borschberg had touched down in Muscat after the first leg of the journey, 13 hours and two minutes after leaving Abu Dhabi.
"The adventure has started," Solar Impulse chairman Bertrand Piccard said just after Borschberg took off in the early morning from Abu Dhabi's Al-Bateen airport on the historic journey aimed at promoting green energy.
Borschberg spoke of an "emotional" trip, telling reporters in Muscat he cruised at 6,000 metres (almost 20,000 feet) because the trip was "short".
Piccard is set to fly higher on the trip to Ahmedabad, Borschberg said.
Both pilots wore traditional Omani turbans as they received an official welcome in the Gulf sultanate.
UN chief Ban Ki-moon hailed the venture and congratulated the pilots.
"We take inspiration from their example and efforts to harness the power of multilateralism to address climate change and to inspire the world to achieve sustainable development," his spokesman said.
"With their daring and determination, we can all fly into a new sustainable future."
Research and testing
Si2's takeoff, which had originally been scheduled for Saturday but was delayed by high winds, capped 13 years of research and testing by the two Swiss pilots.
Live video streaming on the website monitoring the unique aircraft's progress showed the pilot, wearing an orange jumpsuit, breathing using an oxygen mask.
"From Mission Control Center in #Monaco the engineers are helping me to perform Oxygen Mask tests from #SolarImpulse," he tweeted.
Borschberg earlier called his wife from on board, according to the live feed.
The wingspan of the one-seater Si2 is slightly bigger than that of a jumbo jet, but its weight is around that of a family car.
From Muscat, it will make 12 stops on an epic journey spread over five months, with a total flight time of around 25 days.
Later legs will take it to Myanmar, China, Hawaii and New York.
Landings are also earmarked for the midwestern United States and either southern Europe or North Africa, depending on the weather conditions.
The longest single leg will see a lone pilot fly non-stop for five days and nights across the Pacific Ocean between Nanjing, China and Hawaii, a distance of 8,500 kilometres (5,270 miles).
Intensive training
Borschberg and Piccard will alternate flying the plane, which can fly on autopilot during rest breaks.
"You have to make the cockpit like your own house... you go to the toilet, you wash yourself with wet wipes, you eat, you drink, you recline the seat to have some rest, you turn on the autopilot," Piccard told reporters.
Both pilots have undergone intensive training in preparation for the trip, including in yoga and self-hypnosis, allowing them to sleep for periods as short as 20 minutes but still awake feeling refreshed.
All this will happen without burning a drop of fuel.
While in the air, the pilots will be linked to a control centre in Monaco where 65 weathermen, air traffic controllers and engineers will be stationed. A team of 65 ground staff will travel with the two pilots.
"We want to share our vision of a clean future," Piccard said of the mission, which was ridiculed by the aviation industry when it was first unveiled.
But the 57-year-old, who hails from a family of scientist-adventurers and in 1999 became the first person to circumnavigate the globe in a hot air balloon, clung to his belief that clean technology and renewable energy "can achieve the impossible".
The plane is powered by more than 17,000 solar cells built into wings that, at 72 metres, are longer than on a Boeing 747 and approaching those of an Airbus A380 superjumbo.
Thanks to an innovative design, the lightweight carbon fibre aircraft weighs only 2.3 tonnes, about the same as a family 4X4 and less than one percent of the weight of the A380.
The Si2 is the first solar-powered aircraft able to stay aloft for several days and nights.
The propeller-driven craft has four 17.5 horsepower electric motors with rechargeable lithium batteries.
It will travel at 50-100 kilometres per hour, but slower at night to prevent the batteries from draining too quickly.
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