The Prophet (may Allah's peace and blessings be upon him) forbade praying after the Fajr prayer up until the sun rises, and after the ‘Asr prayer until the sun sets
Authentic hadithعن عبد الله بن عباس رضي الله عنه قال: «شَهِد عِندِي رِجَال مَرْضِيُون - وأَرْضَاهُم عِندِي عُمر- أنَّ النَبِي صلى الله عليه وسلم نَهَى عن الصَّلاة بَعد الصُّبح حتَّى تَطلُعَ الشَّمسُ، وبعد العصر حتَّى تَغرُب». وعن أبي سعيد رضي الله عنه عن رسول الله صلى الله عليه وسلم أنه قال: «لا صَلاَة بعد الصُّبح حتَّى تَرتَفِعَ الشَّمسُ، ولا صَلاَة بعد العَصرِ حتَّى تَغِيبَ الشَّمس».
‘Abdullāh ibn ‘Abbās (may Allah be pleased with him) reported: "Some trustworthy men, and the most trustworthy among them in my eyes was ‘Umar, testified before me that the Prophet (may Allah's peace and blessings be upon him) forbade praying after the Fajr prayer up until the sun rises, and after the‘Asr prayer until the sun sets." Abu Sa‘īd (may Allah be pleased with him) reported that the Messenger of Allah (may Allah's peace and blessings be upon him) said: "There should be no prayer after the morning prayer until the sun rises high, and there should be no prayer after the ‘Asr prayer until the sun sets.”
Explanation
In these two Hadīths, the Prophet (may Allah's peace and blessings be upon him) forbade praying after the Fajr prayer up until the sun rises and goes beyond its zenith, as seen by the normal eye for as much as the height of a spear fixed in the horizon. This period is estimated to be between 5 to 15 minutes. He also forbade praying after the ‘Asr prayer up until the sun sets, i.e. few minutes before the Adhān of the Magrib prayer. That is because offering prayer at these two times involves resemblance to the polytheists who worship the sun upon its rising and setting, and Muslims are forbidden from acting like them in their worship, because whoever imitates a people becomes one of them.
Source · Al-Bukhari and Muslim with its two versions
From the Hadith Encyclopedia (hadeethenc.com).