December 25 is the birthday of the Roman Sun God, Sol Invictus.
Dies Natalis Solis Invicti means "the birthday of the unconquered Sun".
The use of the title Sol Invictus allowed several solar deities to be worshipped collectively, including Elah-Gabal, a Syrian sun god, Sol, the god of Emperor Aurelian, and Mithras, a soldiers' god of Persian origin.
Emperor Elagabalus (AD 218-222) introduced the festival, and it reached the height of its popularity under Aurelian, who promoted it as an empire-wide holiday.
The festival was placed on the date of the solstice because this was on this day that the Sun reversed its southward retreat and proved itself to be "unconquered".
These coins are from the emperors Antoninus Pius (AD 138-161), Elagabalus (AD 218-222), Aurelian (AD 270-275), and Constantine (AD 306-337).

Antoninus Pius Provincial AE 23 - Emesa, Syria
Bronze, 23 mm, 8.71 gm, struck AD 138-161 Emesa, catalog: BMC 1
Obverse: Laureate bust right
AVT KAI TI AIL ANTWNEINOC CEB EV
Reverse: Eagle standing right on sacred stone of Elagabal, B in right field
EMICHNWN

Elagabalus Denarius - Sol Advancing
Silver, 19 mm, 1.49 gm, struck AD 221 Rome, catalog: RIC 40
Obverse: Laureate and draped bust right
IMP ANTONINVS PIVS AVG
Reverse: Sol advancing left with raised hand and whip
PM TRP II II COS III PP

Elagabalus Denarius - Sol Standing
Silver, 18 mm, 2.98 gm, struck AD 219 Rome, catalog: RIC 17
Obverse: Laureate and draped bust right
IMP ANTONINVS PIVS AVG
Reverse: Sol standing left, right hand raised, left hand holding whip
PM TRP II COS II PP

Aurelian Antoninianus - Sol Invictus with Captive
Bronze, 22 mm, 3.60 gm, struck AD 274 Serdica (Modern day Sofia Bulgaria), catalog: RIC Serdica 276
Obverse: Radiate and cuirassed bust right
IMP AVRELIANVS AVG
Reverse: Sol standing left, right hand raised, holding globe, captive at foot
ORIENS AVG - Exergue: S (Serdica mint)

Aurelian Antoninianus - Sol Invictus Walking
Bronze, 22 mm, 3.94 gm, struck AD 274-275 Rome, catalog: RIC 64
Obverse: Radiate and cuirassed bust right
IMP AVRELIANVS AVG
Reverse: Sol walking right with laurel-branch and bow, treading down an enemy
ORIENS AVG, letter A to left - Exergue: XXIR (Rome mint)
Catalog: RIC Rome 64

Constantine AE Follis - Sol Invictus
Bronze, 19 mm, 2.64 gm, struck AD 313-314 Ticinum, catalog: RIC Ticinum 8
Obverse: Laureate cuirassed bust
IMP CONSTANTINVS PF AVG
Reverse: Sol standing left with orb and raising right hand
SOLI INVICTO COMITI - Exergue: ST (Ticinum Mint)

Constantine AE Follis - Sol Invictus
Bronze, 21 mm, 2.66 gm, struck around AD 326 Rome, catalog: RIC Rome 2
Obverse: Laureate cuirassed bust
IMP CONSTANTINVS PF AVG
Reverse: Sol standing left with orb and raising right hand, captive to left of Sol
SOLI INVICTO COMITI - Exergue: RP (Rome Mint)
