A spacecraft leaves Earth fast enough to orbit the Sun on its own, coasts for months on a path that meets Mars, then brakes so Mars can capture it.
Mars and Earth both circle the Sun. To get from one to the other, a spacecraft is put on its own orbit around the Sun, one that touches Earth's orbit at one end and Mars's orbit at the other. The cheapest such path is called a Hohmann transfer: it starts at its closest point to the Sun (at Earth) and ends at its farthest (at Mars).
Mars has to arrive at the far end of the path at the same time as the spacecraft. So the planets must be in the right line-up at launch. NASA notes that the chance to launch on a minimum-energy path to Mars comes about every 25 months.
Mangalyaan, the Mars Orbiter Mission, was launched by PSLV-C25 from Sriharikota on 5 November 2013 into a parking orbit around Earth.
On 1 December 2013 its trans-Mars injection burn sent it out of Earth's orbit, on a path to meet Mars on 24 September 2014.
On that day the spacecraft fired its 440 newton main engine, together with eight smaller engines, to slow down. Mars captured it into an orbit from 421.7 km to 76,993.6 km above the planet, taking about 72 hours 51 minutes per lap. ISRO became the fourth space agency to put a spacecraft in Mars orbit.
The mission was designed to last six months. It was still working six years later, in September 2020.
A spacecraft coasting in on its orbit around the Sun arrives too fast for Mars to hold it. NASA notes that getting into orbit needs a braking burn, and that landers can also use the thin air, parachutes and rocket burns to slow down.
Mangalyaan's braking burn began at 07:17:32 IST on 24 September 2014. By the end of that morning the spacecraft was in an elliptical orbit around Mars.
Find the gap between the 1 December 2013 trans-Mars injection and the 24 September 2014 arrival: almost ten months of coasting around the Sun. Explore the Mangalyaan timeline
For a Hohmann transfer between circular orbits of radius r₁ (Earth) and r₂ (Mars) around the Sun:
a_t = (r₁ + r₂)/2, transfer time = π√(a_t³ / GM_Sun)
Worked numbers (NASA fact sheets): r₁ = 149.598 million km, r₂ = 227.956 million km, GM_Sun = 1.32712 × 10¹¹ km³/s².
Mangalyaan's coast, from 1 December 2013 to 24 September 2014, was 297 days. That is longer than the idealised 259 days, because real planetary orbits are not perfect circles in one plane and a real trajectory is chosen for the actual dates.
The ~25-month launch rhythm is the synodic period: the time for Earth to lap Mars again. NASA's Mars fact sheet gives 779.94 days, about 25.6 months. Equivalently 1/S = 1/365.256 − 1/686.980, which gives S ≈ 780 days.
The most fuel-efficient transfer between two circular orbits: an ellipse touching the inner orbit at its closest point and the outer orbit at its farthest point.
The spacecraft must reach Mars's orbit just when Mars is there. The right Earth–Mars line-up comes round about every 25 to 26 months, the synodic period.
Its trans-Mars injection was on 1 December 2013, and it entered Mars orbit on 24 September 2014.